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IN  MEMORIAM 
FLORIAN  CAJORl 


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PRINCIPLES  OF  ELEMENTARY  ALGEBRA 


♦Th^>C^o. 


THE  PEINCIPLES 


ELEMENTARY   ALGEBRA 


BY 


N.   F.    pUPUIS,   M.A.,   F.R.S.C. 

Protessor  op  Pure  Mathematics  in  the  University  of  Queen's 
College,  Kingston,  Canada 


BEYISKD^AITD    O^EnEC'iJ^D   ^VITION' 


Wehj  got* 
THE    MACMILLAN   COMPANY 

LONDON :   MACMILLAN  &  CO.,  Ltd. 
1900 

All  rights  reserved 


'S 


Copyright,  1892, 
By  MACMILLAX  AND  CO. 


Set  up  and  clectrotyped  August,  1892.     Reprinted  September, 
1900. 


'y:■^M^w^^.:^'\^ 


J.  S.  Cashing  &  Co.  —  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


PREFACE. 


In  the  following  pages  I  have  endeavored  to  put  into 
form  what  in  my  opinion  should  constitute  an  Inter- 
mediate Algebra,  intermediate  in  the  sense  that  it  is  not 
intended  for  absolute  beginners,  nor  yet  for  the  accom- 
plished algebraist,  but  as  a  stepping-stone  to  assist  the 
student  in  passing  from  the  former  stage  to  the  latter. 

The  work  covers  pretty  well  the  whole  range  of 
elementary  algebraic  subjects,  and  in  the  treatment  of 
these  subjects  fundamental  principles  and  clear  ideas  are 
considered  as  of  more  importance  than  mere  mechanical 
processes.  The  treatment,  especially  in  the  higher 
parts,  is  not  exhaustive ;  but  it  i^  hoped  that  the  treat- 
ment is  sufficiently  full  to  enable  the  reader  who  has 
mastered  the  work  as  here  presented,  to  take  up  with 
profit  special  treatises  upon  the  various  subjects. 

Much  prominence  is  given  to  the  formal  laws  of 
Algebra  and  to  the  subject  of  factoring,  and  the  theory 
of  the  solution  of  the  quadratic  and  other  equations  is 
deduced  from  the  principles  of  factorization. 

The  Sigma  notation  is  introduced  early  in  the  course, 
as  being  easily  understood,  and  of  great  value  in  writ- 
ill 


91129? 


iv  PREFACE. 

ing  and  remembering  important  symmetrical  algebraic 
forms. 

Synthetic  Division  is  commonly  employed,  and  the 
principles  of  its  operation  are  extended  to  the  finding 
of  the  highest  common  factor. 

Except  in  the  case  of  surds,  no  special  method  is 
given  for  finding  the  square  root  of  an  expression  which 
is  a  complete  square,  as  the  operation  is  only  a  case  of 
factoring,  and  a  simple  case  at  that.  For  expressions 
which  are  not  complete  squares,  the  most  rational  method 
is  by  means  of  the  binomial  theorem  or  of  undetermined 
coefficients,  both  of  which  are  amply  dealt  with. 

Probably  the  most  distinctive  feature  of  the  work  is 
the  importance  attached  to  the  interpretation  of  alge- 
braic expressions  and  results.  Algebra  is  an  unspoken 
language  written  in  symbols,  of  which  the  manipulation 
is  largely  a  matter  of  mechanical  method  and  of  the 
observance  of  certain  rules  of  operation.  The  results 
arrived  at  have  little  interest  and  no  special  meaning 
until  they  are  interpreted.  This  interpretation  is  either 
Arithmetical,  that  is,  into  ideas  involving  numbers  and 
the  operations  performed  upon  numbers ;  or  Geometri- 
cal, that  is,  into  ideas  concerning  magnitudes  and  their 
relations.  Both  interpretations  necessitate  observation 
and  the  exercise  of  thought;  but  the  geometrical  offers 
the  wider  scope  for  ingenuity,  and  is  the  better  test 
of  mathematical  ability.     In  several  cases,  as  in  that 


PREFACE.  V 

of  the  quadratic  equation,  the  solution  frequently  gets 
its  complete  explanation  only  through  its  geometric 
interpretation.  Hence  geometrical  problems  are  freely 
introduced,  and  the  relations  between  the  symbolism  of 
Algebra  and  the  fundamental  ideas  of  Geometry  are 
discussed  at  some  length. 

The  Graph  is  freely  employed  both  as  a  means  of 
illustration  and  as  a  medium  of  independent  research; 
and  through  these  means  an  effort  is  made  to  connect 
Algebra  with  Arithmetic  upon  the  one  hand,  and  with 
Geometry  upon  the  other. 

The  exercises  are  numerous  and  varied,  and  I  trust 
that  they  will  be  found  to  be  fairly  free  from  errors. 

N.  F.  D. 


CONTENTS. 


Ohaptbb  Page 

I.  Symbols,  Definitions,  and  Formal  Laws 1 

II.   The  Four  Elementary  Operations 17 

III.  Factors  and  Factorization 45 

IV.  Highest  Common  Factor.  Least  Common  Multiple  64 
V.  Fractions.    c»  and  0 74 

VI.  Ratio,    Proportion,   Variation   or   Generalized 

Proportion 90 

VII.   Indices  and  Surds 100 

VIII.   Concrete   Quantity.     Geometrical   Interpreta- 
tions.    The  Graph 114 

IX.   The  Quadratic 136 

X.   Indeterminate  and  Simultaneous  Equations  of 

THE  First  Degree.     Simultaneous  Quadratics.  163 
XI.   Remainder  Theorem.     Transformation  of  Func- 
tions.    Approximation  to  Roots 176 

XII.   Progressions.     Interest  and  Annuities 188 

XIII.  Permutations,  Combinations,  Binomial  Theorem  210 

XIV.  Inequalities 227 

XV.   Undetermined  Coefficients  and  their  Applica- 
tions      231 

XVI.    Continued  Fractions 244 

XVII.    Logarithms  and  Exponentials 255 

XVIII.    Series  and  Interpolation 274 

XIX.    Elementary  Determinants 296 

For  a  more  detailed  statement  consult  the  Index  at  the  end  of  the  book. 

vii 


CHAPTER  I. 
Symbols,  Definitions,  and  Formal  Laws. 

1.  Arithmetic  is  pure  or  concrete.  Pure  arithmetic 
deals  with  abstract  number  or  numerical  quantity.  Con- 
crete arithmetic  has  relation  to  numbers  of  concrete 
objects  or  things. 

Thus  3  is  an  abstract  number,  but  3  days  is  concrete. 

Algebra  is  primarily  related  to  pure  arithmetic,  but  its 
extension  to  concrete  arithmetic  is  an  easy  matter. 

The  quantities  which  are  the  subject  of  arithmetic  are 
of  three  kinds : 

(1)  Whole  numbers  or  integers  ; 

(2)  Symbolized  operations  called  fractions ; 

(3)  Numerical  quantities  which  cannot  be  exactly 
expressed  as  integers  or  fractions,  but  whose  values  may 
be  expressed  to  any  required  degree  of  approximation. 
Such  are  the  square  roots  of  the  non-square  numbers, 
the  cube  roots  of  the  non-cube  numbers,  etc.  This  third 
class  goes  under  the  general  name  of  incommensurables. 

The  expression  numerical  quantity,  and  frequently  the 
word  number,  will  be  taken  to  denote  any  of  the  three 
classes. 

2.  Numbers  are  fundamentally  subject  to  two  opera- 
tions —  increase  and  diminution ;  but  convenience,  drawn 
from  experience,  has  led  us  to  enumerate  four  elementary 

1 


2       SYMBOLS,    DEFINITIONS,    AND   FORMAL   LAWS. 

operations,  viz. :   Addition,  Subtraction,   Multiplication, 
and  Division. 

All  higher  operations  on  numbers  are  but  combinations 
of  the  four  elementary  ones. 

3.  Algebra  originated  in  arithmetic,  and  elementary 
algebra  is  arithmetic  generalized,  the  generalization  being 
effected  by  employing  symbols,  usually  non-numerical, 
to  stand  for  and  represent  not  only  numbers  or  numeri- 
cal quantities,  but  also  the  operations  usually  performed 
upon  numbers. 

Thus  algebra  becomes  a  symbolic  language  in  which 
numbers  and  the  operations  upon  them  are  written. 

The  symbols  of  algebra  are  thus  primarily  of  two 
kinds : 

(1)  Quantitative  symbols,  which  represent  numerical 
quantities,  and 

(2)  Operative  symbols,  which  indicate  operations  to 
be  performed  upon  the  quantity  denoted  by  the  quantita- 
tive symbol. 

A  third  class,  called  verbal  symbols,  may  be  enumerated, 
in  which  the  symbol  is  a  convenient  contraction  for  a 
word  or  phrase. 

4.  The  quantitative  symbols  are  usually  letters.  The 
operative  symbols,  especially  in  elementary  algebra  and 
in  arithmetic,  are  mostly  marks  or  signs  which  are  not 
letters.  Relative  position  is  employed  to  denote  some 
operations,  and  in  higher  algebra  very  complex  operations 
are  often  denoted  by  letters. 

The  verbal  symbols  do  not  denote  quantity,  and  they 
cannot  be  said,  in  general,  to  denote  operations. 


SYMBOLS,   DEFINITIONS,   AND  FOEMAL   LAWS.       3 

The  principal  verbal  symbols  are : 

(1)  =  and  =,  either  of  which  denotes  that  all  that 
precedes  the  symbol,  taken  in  its  totality,  is  equal  to  or 
is  the  same  as  all  that  follows  the  symbol,  taken  also  in 
its  totality. 

(2)  >  and  <.  The  first  denotes  that  all  that  pre- 
cedes the  symbol,  taken  in  its  totality,  is  greater  than  all 
that  follows  the  symbol,  taken  in  its  totality;  and  the 
second  is  like  the  first  with  less  put  for  greater. 

Other  verbal  symbols  will  be  introduced  as  required. 

5.  From  Art.  3  it  is  seen  that  operations  in  arithmetic 
must  be  special  cases  of  more  general  operations  in 
algebra.  And  hence  it  follows  that  arithmetic  and 
algebra  must  proceed  on  similar  principles,  and  must  be 
subject  to  the  same  formal  operative  laws. 

That  the  generalizing  process  of  algebra  should  intro- 
duce new  ideas  into  arithmetic  is  to  be  expected;  and 
that  this  generalization  should  carry  us  beyond  the  neces- 
sarily limited  field  of  arithmetic  is  also  to  be  expected. 
Illustrations  will  occur  hereafter. 

6.  The  operative  symbol  -f  (plus)  denotes  addition, 
and  tells  us  that  the  quantity  before  which  it  stands,  and 
to  which  it  belongs,  is  to  be  added  to  whatever  precedes. 

Thus,  5  +  3  tells  us  that  3  is  to  be  added  to  5,  and 
0  -f-  3  is  the  same  as  the  arithmetical  number  3. 

Similarly,  +  a  is  the  same  as  a,  whatever  a  stands  for ; 
and  for  this  reason  the  sign  +  is  seldom  written  when- 
ever it  can  be  dispensed  with  without  producing  ambiguity. 

a  -f  5  is  the  same  as   -j-  a  +  6,  and  indicates  that  the 


4       SYMBOLS,   DEFINITIONS,    AND   FORMAL   LAWS. 

number  denoted  by  h  is   to  be   added  to   the   number 
denoted  by  a. 

7.  Any  interpretable  combination  of  quantitative  and 
operative  symbols  is  an  algebraic  expression.  We  shall, 
in  the  meantime,  confine  ourselves  to  expressions  written 
in  a  single  line,  as 

3a6-f  2c-f  d-,  etc. 

In  arithmetic  we  know  that  3  +  5  is  in  its  sum  the 
same  as  5-J-3,  and  3  +  5  +  8  is  the  same  as  3  +  8  +  5, 
the  same  as  8  +  5  +  3,  etc.  And  as  this  must  be  a  par- 
ticular case  of  algebra  (Art.  5),  we  must  have  a  +  6  = 
h-\-a,  a-\-'b-\-c  =  a-\-'c-{-h  =  h-^c-\-a  =  etc. 

This  is  the  Commutative  Laiv  for  Addition,  and  is  ex- 
pressed by  saying  that  the  order  of  adding  quantities  is 
arbitrary^  or  the  sum  is  independent  of  the  order  of  the 
addends. 

8.  The  symbol  —  (minus)  placed  before  a  quantity 
indicates  that  the  quantity  is  to  be  subtracted  from 
whatever  precedes  the  symbol. 

Thus,  5  —  3  tells  us  that  3  is  to  be  subtracted  from  5 ; 
and  a  —  b  tells  us  that  the  quantity  denoted  by  b  is  to 
be  subtracted  from  that  denoted  by  a.  Now,  a  and  b 
denoting  any  numerical  quantities,  as  long  as  a  is  greater 
than  b  the  subtraction  is  arithmetically  possible,  and  the 
result  is  an  arithmetical  quantity.  But  if  a  is  less  than 
b,  the  operation  symbolized  is  not  arithmetically  possible. 
The  expression  a  —  &  is  then  a  symbolic  representation 
of  an  operation  that  cannot  be  arithmetically  performed, 
and  the  result  of  the  operation,  whatever  it  may  be,  is 
not  arithmetical. 


SYMBOLS,    DEFINITIONS,    AND   FORMAL   LAWS.       5 

Thus,  0  —  3,  which  is  simply  written  —  3,  and  which 
is  called  a  negative  number,  does  not  belong  to  pure 
arithmetic,  but  is  an  idea  introduced  into  algebraic  arith- 
metic by  generalizing  the  operation  of  subtraction. 

And  thus  to  every  pure  number,  called  now  a  positive 
number,  corresponds  an  algebraical  negative  number,  the 
relation  between  corresponding  numbers  being  that  their 
algebraic  sum  is  zero  or  nothing. 

Negative  numbers  are  important  in  their  relations  to 
concrete  arithmetic,  and  especially  where  geometric  ideas 
are  concerned.  This  matter  will  be  dealt  with  in  Chap- 
ter VIII. 

9.  It  is  said  in  Art.  2  that  numbers  are  fundamentally 
capable  of  only  increase  or  diminution.  Hence  +  and 
—  symbolize  the  two  great  operations  in  arithmetic  and 
algebra.     These  are  distinctively  the  signs  of  algebra. 

By  the  sign  of  a  quantity  is  meant  that  one  of  these 
two  signs  which  precedes  the  quantity;  and  to  change 
signs  is  to  change  +  to  —  and  —  to  +  throughout. 

Also,  two  quantities  have  like  signs  when  both  are 
preceded  by  +  or  both  by  — ;  otherwise  they  have 
unlike  signs. 

When  no  sign  is  written,  -f-  is  understood. 

10.  That  part  of  an  expression  included  between  two 
consecutive  signs  is  called  a  term. 

To  indicate  that  any  portion  of  an  expression  lying 
between  two  non-consecutive  signs  is  to  be  taken  in  its 
totality  as  a  single  term,  we  enclose  the  portion  within 
brackets. 

Thus,  in  the  expression  a-f26c  —  4(3c  +  2  ab),  a,  2  be 


6       SYMBOLS,    DEFINITIONS,    AND   FORMAL   LAWS. 

are  single  or  simple  terms,  and  3c-{-2ab  is  to  be  con- 
sidered as  one  complex  term. 

The  sign  —  and  the  number  4  preceding  the  brackets, 
apply  to  its  contents  in  their  totality. 

Instead  of  brackets,  we  often  employ  a  line  called  a 
vinculum  J  drawn,  above  the  portion  indicated,  as,  3  c  -j- 1?  a6 ; 
and  sometimes,  for  special  reasons,  this  line  is  placed 
beneath  instead  of  above. 

11.  The  symbol  x  or  •  (into  or  by)  indicates  that  the 
quantity  following  the  symbol  is  to  act  as  a  multiplier 
upon  the  quantity  preceding  the  symbol. 

With  numerical  symbols,  as  4,  7,  etc.,  it  is  evident  that 
we  cannot  dispense  with  the  symbol,  as  34  is  not  the 
same  as  3x4;  but  this  difficulty  does  not  exist  with 
letters,  and  hence  we  usually  write  ab  instead  of  a  x  6 
or  a  •  &. 

In  this  case  relative  position  or  juxtaposition  becomes 
a  symbol  of  multiplication. 

12.  The  parts  which  make  up  a  term,  or  an  expression, 
by  multiplication  only,  are  factors  of  the  term  or  expres- 
sion. 

Thus,  3,  a,  &,  and  c  are  factors  of  Sabc,  3  being  a 
numerical  factor,  and  a,  b,  c  literal  factors.  So,  also, 
4,  a,  b  -\-  c,  and  c-^  a  are  factors  of  the  expression 
4a(6  + c)(c  +  a). 

13.  In  arithmetic  we  know  that  4x6  is  the  same  in 
value  as  6x4;  3x2x5  is  the  same  as  2x5x3,  etc. ; 
and  as  this  must  be  a  particular  case  of  algebra,  we  must 
agree  that  ab  =  ba,  that  abc  =  bca  =  etc. 

This  is  the  Commutative  Law  for  Multiplication ,  and  is 


SYMBOLS,   DEFINITIONS,    AND   FORMAL   LAWS.       7 

expressed  by  saying  that  a  product  is  indepefident  of  the 
order  of  its  factors. 

Thus,  ab  means,  indifferently,  that  b  multiplies  a  or 
that  a  multiplies  b. 

14.  Since  multiplication  by  +  1  effects  no  change  in 
the  quantity  multiplied,  we  have 

(+l)(+a)  =  +  a; 

Or,  +  multiplied  by  +  gives  +  in  the  product. 

Again,  a  —  a==0  =  a  +  (—  a)  =  a4-(+l)(—  a),as  —  a 
may  be  taken  in  its  totality  by  placing  it  within  brackets. 

Hence  (+ 1)  (— a)  must  be  —a; 

Or,  +  multiplied  by  —  gives  — ,  and,  by  the  commuta- 
tive law  for  multiplication,  —  multiplied  by  +  gives  — 
in  the  product. 

Again,  a  ~  a  =  0, 

and  writing  —  b  for  a,  we  have 

-b-{-b)  =  0; 
and  hence  —(—6), or  (  —  !)(  —  &)  must  be  the  same  as  +6; 

Or,  —  multiplied  by  —  gives  +  in  the  product. 

Collecting  results,  we  have  as  the  Law  of  Signs :  The 
multiplication  of  two  like  signs  gives  +,  and  the  multi- 
plication of  two  unlike  signs  gives  —  in  the  product. 

15.  It  is  readily  established  in  arithmetic  that 

3  (4 +  2  + 5)  =  3x4  +  3x2  +  3x5. 

And  as  this  must  be  a  particular  case  of  algebra,  we 
must  assume  that 

a  (b  -{- c -{- d)  =  ab  +  ac -\-  ad. 


8       SYMBOLS,    DEFINITIONS,    AND   FORMAL   LAWS. 

We  have  here  three  terms,  b,  c,  and  d,  which  are  placed 
in  brackets  and  taken  as  a  complex  term,  and  we  have 
a  multiplier  a  which  operates  upon  this  complex  term. 
And  we  see  that  we  may  distribute  this  operator  so  as 
to  act  separately  upon  each  of  the  terms  of  which  the 
complex  term  is  composed. 

This  is  the  Distributive  Law  for  multiplication. 

Some  other  operations  — like  multiplication  —  are  dis- 
tributive, and  some  are  not.  The  case  for  each  operation 
must  be  worked  out  and  learned  by  itself. 

16.  A  term,  such  as  aaabbc,  containing  repeated  letters, 
is  simplified  in  form  by  writing  it  a^b\  in  which  the 
small  numerals  placed  to  the  right  and  above  a  and  b 
show  how  many  times  each  of  these  letters  enters, 
respectively,  as  a  factor. 

The  symbol  a^  is  read  ^a  cubed,'  and  b^  is  read  ^b 
squared.' 

The  letters  a  and  b  are  subjects  or  roots,  and  the  3  and 
2  are  exponents  or  indices. 

Similarly,  a"",  where  n  denotes  any  integer,  is  the  nth. 
power  of  a,  read  ^  a  nth-power,'  or  ^  a-to-the-nth/  and 
denotes  that  a  is  to  be  taken  n  times  as  a  factor. 

Now  if  n=p-{-q,  i.e.  if  n  be  separated  into  two 
integers  denoted  by  p  and  g,  we  have 

a**=  a  •  a  •  a  •••  to  n  factors. 
Or  a^+*  =  a  •  a  •  a  •  •  •  to  jp  factors  x  a  •  a  •  a  •  •  •  to  g  factors 
=  aP .  a^. 

This  expresses  the  Index  Law,  its  statement  being  that 
the  product  of  any  powers  of  the  same  subject  is  that  power 
of  the  subject  ivhich  is  denoted  by  the  sum  of  the  exponents. 


SYMBOLS,   DEFINITIONS,    AND   FORMAL  LAWS.       9 

Thus,  a'^ '  a^  z=  a^  'y  a^  -  a^  -  a^  =  a^^,  etc. 

Of  course  a  is  the  same  as  a\ 

17.  The  Commutative  laws  for  addition  and  multipli- 
cation, the  Distributive  law  for  multiplication,  and  the 
Index  law  are  the  great  formal  laws  of  elementary- 
algebra,  and  its  symbolism,  its  principles  of  operation, 
and  its  results  are  applicable  to  any  subject  which  by 
any  consistent  process  of  interpretation  can  be  shown  to 
be  governed  by  these  laws. 

A  proper  conception  of  this  fact  opens  the  way  to 
important  extensions  in  the  applications  of  algebra. 

The  foregoing  laws  belong  to  elementary  algebra 
because  this  subject  is  a  generalization  of  arithmetic,  in 
which  these  laws  hold ;  but  that  they  are  not  all  essential 
to  all  kinds  of  algebra,  we  know,  as  we  have  a  special 
algebra.  Quaternions,  in  which  the  commutative  law  for 
multiplication  does  not  in  general  hold  true.  But  this 
latter  algebra  is  not  generalized  arithmetic. 

18.  A  single  letter  as  a  quantitative  symbol  is  said  to 
be  of  one  dimension,  and  the  number  of  dimensions  of  a 
term,  which  consists  of  multiplications  only,  is  the  num- 
ber of  letters,  either  expressed  or  implied,  which  the 
term  contains. 

Thus,  ahCy  a^b,  a^  are  each  of  three  dimensions;  and 
3  a^bcdy  4  a^b%  a^b^c  are  each  of  five  dimensions,  a  numerical 
factor,  as  3  or  4,  having  no  dimensions. 

The  number  of  dimensions  of  a  term  constitutes  its 
degree  ;  thus  the  first  three  of  the  preceding  terms  are 
of  the  third  degree,  and  the  last  three  of  the  fifth  degree. 

Usually  a  term  is  said  to  be  of  a  certain  degree  in 
some  particular  letter  or  letters. 


10      SYMBOLS,   DEFINITIONS,    AND   FORMAL   LAWS. 

Tliiis,  Sab^x^  is  of  the  first  degree  in  a,  of  the  second 
in  b,  and  of  the  third  in  x. 

When  the  degree  of  each  term  of  an  expression  has 
reference  to  the  same  particular  letter,  this  letter  is 
called  a  variable,  and  any  other  letters  occurring  in  the 
expression  are  constants. 

Thus,  in  the  expression  ax^  -^bx  -{-  c,  ax^  is  of  the  sec- 
ond degree  in  x,  bx  is  of  the  first,  and  c,  not  containing 
Xj  is  the  absolute  or  independent  term.  In  this  case  x  is 
the  variable,  and  a,  b,  c  are  constants. 

The  term  dimension  is  derived  from  geometry,  and  its 
significance  will  be  more  fully  seen  hereafter. 

19.  An  expression  of  one  dimension  in  each  term  is  a 
linear  expression,  as  a  +  ^  +  <^- 

An  expression  which  contains  a  variable  in  the  first 
degree  only  is  linear  in  that  variable. 

Thus,  ^abx  is  linear  in  x,  and  3a?>  is  the  coefficient  of  x. 

Similarly,  x  —  a  and  ax  +  be  are  both  linear  in  x, 
although  the  first  is  of  one  dimension,  and  the  second  of 
two,  in  regard  to  all  the  letters. 

An  expression  which  is  of  the  same  dimensions  in 
every  term  is  homogeneous.  Such  expressions  are  specially 
important. 

Thus,  a  +  6  4-  c,  ab  -\- be -[-  ca,  a^  -f  3  a^&  -f  3  ab'^  -f  b^  are 
each  homogeneous  with  respect  to  all  the  letters. 

20.  A  quantitative  symbol  stands  for  any  numerical 
quantity  whatever,  and  operations  upon  such  general 
quantities  can  be  only  symbolically  indicated.  Thus  a 
and  b  being  any  quantities,  we  denote  their  sum  by 
a  +  6,  and  their  product  by  ab. 


SYMBOLS,   DEFINITIONS,    AND   FORMAL   LAWS.      11 

Herein  lies  the  advantage  of  algebraic  symbolization, 
its  great  powers  being  due  to  two  things : 

(1)  The  universal  significance  of  the  quantitative 
symbol,  and 

(2)  That  the  operations  performed,  unlike  those  in 
arithmetic,  are  not  lost  sight  of,  so  that  a  chain  of  con- 
secutive operations  may  be  so  reduced  by  transformations, 
as  to  depend  upon  the  smallest  possible  cycle  of  such 
operations. 

21.  As  algebra  is  generalized  arithmetic,  every  alge- 
braic relation,  which  is  arithmetically  interpretable,  ex- 
presses some  general  relation  amongst  numbers. 

Thus,  ah  (a  +  h)  gives  a%  +  ab^  by  distributing  ah,  or 
ah  {a  -^  h)  =  a^h -{-  ab\ 

This  interpreted  gives  the  arithmetical  theorem : 

The  sum  of  two  numbers  multiplied  by  their  product 
is  equal  to  the  sum  of  the  numbers  formed  by  multiply- 
ing each  number  by  the  square  of  the  other. 

It  may  be  remarked  that  theorems  like  the  foregoing 
cannot,  in  general,  be  proved  by  an  arithmetical  process. 
Repeated  trials  with  different  numbers  would  give  a  sort 
of  moral  proof,  but  not  a  mathematical  one,  since  we 
could  not  possibly  try  all  numbers.  On  the  other  hand, 
the  quantitative  symbol  standing  at  once  for  any,  and 
hence  for  every  number,  gives  a  proof  which  is  both 
rigid  and  universal. 

EXERCISE  I.  a. 

1.  The  following  are  identities  arising  from  distribution  ;  inter- 
pret them  as  arithmetical  theorems. 

i.   a(a+  b)  =  a'^  -^  ah.  ii.   ab(a  —  b)  =  a^h  —  a6"^. 


12      SYMBOLS,   DEFINITIONS,    AND   FORMAL   LAWS. 

iii.  (a-^b)(a-b)  =  a^-b^.  y.  (a-b)(a-b)=a-\-b^-2ah. 

iv.  (a+6)(a+6)=a2+62+2a&.    yI  (a -{- b)  +  (a  -  b)  =  2a. 
vii.   (a  +  5)  -(«-&)  =  2  5. 
viii.   (a  +  by  +  (a  -  6)2  =  2(a2  +  52). 
ix.   (a  +  6)2-(a-6)2  =  4a6. 
X.  (a  +  6  +  c)2  =  a2  _|_  52  +  ^2  4  2(a&  +  6c  +  ca). 

2.   Reduce  to  a  single  number  — 


i.   l-{-2(-l  +  l-2)}. 
ii.   3{4  -  5(6  -  7[8  -  9])}. 


iii.  ia-ia-Ki-i-TV-iV])}. 

3.   Condense  as  much  as  possible  — 


i.  2a  —  {Sa—(a  —  b  —  a)}. 


ii.  a-b{l  -6(1  -a- 1-6)}. 

4.  Distribute  the  following  — 

i.  {(a-6)-2(6~c)}.{(a  +  6)+2(6  +  c)}. 

ii.  {(m  +  l)a  +  (n  +  1)6}  •  {(m  -  l)a  +  (n  -  1)6} 

4-  {(m  +  l)a  -  (w  +  1)6} .  {(m  -  l)a  -  (w  -  1)6}- 

5.  If  Ti  is  an  integer,  2  n  is  an  even  integer,  and  2  n  +  1  is  an 
odd  integer. 

6.  The  product  of  two  odd  integers  is  an  odd  integer. 

7.  The  sum  of  two  odd  integers  is  an  even  integer. 

8.  The  square  of  an  odd  integer  is  an  odd  integer. 

9.  The  square  of  an  even  integer  is  divisible  by  4. 

10.  What  power  of  2  is  2»  x  22  x  2i-«  x  2  ? 

11.  What  power  of  a  is  a'"-^  -  aP  -  a^^-p  •  a^-"*  ? 

12.  According  to  the  index  law  a*  x  a~^  =  a*"i  =  a^. 
Hence  interpret  a~\ 


SYMBOLS,    DEFINITIONS,    AND   FORMAL   LAWS.      13 

22.  The  symbols  =  and  =.  The  symbol  =  placed 
between  two  expressions  denotes  that  one  of  the  expres- 
sions may  be  transformed  into  the  other  by  the  formal 
operations  of  algebra.  The  whole  is  then  called  an 
Identical  equation,  or  an  Identity,  and  the  connected 
expressions  are  the  members  of  the  identity. 

Thus,  ah  (a  —  b)  =  a^b  —  ab^  is  an  identity,  since  the 
left-hand  member  is  transformed  into  the  right-hand  one 
by  distribution. 

The  symbol  =  between  two  expressions  tells  us  that 
the  expressions  are  to  be  equal  in  their  totalities,  although, 
in  general,  no  transformation  can  change  one  into  the 
other. 

That  this  condition  of  equality  may  exist,  some  rela- 
tion must  hold  amongst  the  quantitative  symbols,  and, 
usually,  one  of  these,  called  the  variable,  is  to  have  its 
value  so  adjusted  as  to  bring  about  an  identity. 

Thus,  4a  +  a?  —  3=6a  —  1  is  an  equation,  or  a  con- 
ditional  equation,  which  is  true  on  the  condition  that  x 
takes  such  a  value  as  will  make  the  whole  an  identity. 
We  readily  see  that  if  x  stands  for,  or  is  replaced  by 
2a-f  2,  the  condition  is  satisfied.  We  say  then  that 
2a +  2  is  the  value  of  x,  and  that  x  is  the  variable  of  the 
equation. 

The  variable  is  often  called  the  unknown,  and  it  is 
manifest  that  any  letter  occurring  in  the  equation  may 
be  taken  as  the  variable.  If  a  be  so  taken,  its  value  is 
found  to  be  "I  a;  —  1. 

23.  Evidently  an  identity  is  not  affected  by  perform- 
ing the  same  operation  upon  each  of  its  members ;  and, 
as  a  conditional  equation  is  to  be  brought  to  an  identity, 


14      SYMBOLS,   DEFINITIONS,    AND   FORMAL   LAWS. 

the  relation  existing  amongst  its  quantitative  symbols  is 
not  changed  by  performing  the  same  operations  upon 
each  member. 

Thus  in  the  equation  3aj  —  2a4-l==a?  +  a  —  4,  we  may 
subtract  x  from  each  side,  add  2  a  to  each  side,  and  sub- 
tract 1  from  each  side.     We  then  have 

2a;  =  3a-5. 

Then  dividing  each  side  by  2,  we  get  as  the  value  of  x, 

We  notice  that  we  transfer  any  term  from  one  member 
to  the  other  by  writing  it  with  a  changed  sign  in  the 
other. 

Thus,  x-{-a  =  b  gives  x  =  b  —  a. 

The  determination  of  the  value  of  the  variable,  in  terms 
of  the  constants  of  an  equation,  is  called  the  solution  of 
the  equation.  And  although  the  solution  of  equations 
does  not  constitute  the  whole  of  algebra,  it  undoubtedly 
forms  a  very  important  part  of  it. 

The  following  examples  are  given  by  way  of  illus- 
tration : 

Ex.  1.   To  find  the  value  of  x  in  the  equation 

3{a  -  4(1  -  x)}  =  2{a  +  3(  a:- 1)}. 

Performing  all  the  distributions, 

Sa-12-{-12x=z2a-\-6x-6. 

Transferring  6  x  from  right  to  left,  and  3  a  —  12  from  left  to 
riglit,  we  have 

6x  =  6  -a. 

Dividing  by  6,  x  =  1  —  la. 


SYMBOLS,    DEFINITIONS,    AND   FORMAL   LAWS.      15 

Ex.  2.  To  find  a  number  which  exceeds  the  sum  of  its  third  and 
fourth  parts  by  10. 

Let  X  denote  the  number;  the  statement  of  the  problem  is  alge- 
braically expressed  as 

\x-\-\x  —  x  —  \^. 

Multiplying  by  12,    4  x  +  3  x  =  12  x  -  120, 

whence  x  =  24. 

Ex.  3.  A  lends  to  B  one  third  of  a  dollar  more  than  ^  of  his 
money,  and  to  C  one  half  a  dollar  more  than  J  of  what  he  has  left. 
A  has  then  $  6.  How  much  had  he  at  first,  and  how  much  did  he 
lend  to  B,  and  to  C  ? 

Let  x  denote  A's  money  at  first. 

He  lends  to  B,  J  x  +  J  dollars. 

He  has  left,  x  —  Qx  +  i),  or  f  x  —  |  dollars. 

He  lends  to  C,  2  (§  ^  —  i)  +  i'  or  ix  +  i  dollars. 

He  has  left,  (f  ^  —  i)  —  ( 3  ic  +  |^) ,  or  |  x  —  f  dollars  ; 

and  this  is  $6. 

Whence  x  =  20  ;  and  B's  loan  =  C's  loan  =  $  7. 

EXERCISE  I.  b. 

1.  Prove  the  following  identities  — 
i.  (a  +  5)2  =  a2-f  h'^  +  2ab. 

ii.  (a2  +  52)  (c2  +  ^2)  =  ^ac  +  hdy  +  {ad  -  hey. 

ill.  (a2  _  52)(<o2  _  (^2)  ^  (^c  -  hdy  -  {ad  -  bey 

=  {ac-j-bdy-{ad-{-  bey. 
iv.   (a  +  6  +  cy+{a  +  b  -  cy  -^(b  -{-  c  -  ay  +{c  +  a  -  by 

=  4(a2-|_  52_^c2). 
V.  (a2  4.  ^,2)2  =  (^2  _  52)2  _|_  (2  aby. 


16      SYMBOLS,   DEFINITIONS,    AND   FORMAL   LAWS. 

2.  Interpret  ii.  and  iii.  of  I  as  theorems  in  numbers. 

3.  By  means  of  v.  of  1  find  two  numbers  such  that  the  sum  of 
their  squares  shall  be  a  square.     Make  a  table  of  such  numbers. 

4.  Find  the  value  of  x  in  each  of  the  following  — 

i.        x-\-ix-hix  =  2x  —  2. 
ii.    3(l-2.3"^^)=:2{l  +  2(.x-2)}. 
6.   Simplify  l-{l-(l-x)}+l  +  {l-(l  +  a^)}+a:-{x-(x-l)}.    . 

6.  Find  x  in  the  equation  {x  —  4)  (a:  +  6)  =  (x  +  8)  (x  +  2). 

7.  Find  a  number  whose  half  exceeds  the  sum  of  its  fourth  and 
fifth  parts  by  40. 

8.  Find  a  number  such  that  if  it  be  increased  by  a,  and  if  it  be 
diminished  by  b,  one  third  of  the  first  result  is  equal  to  one  half 
the  second. 

9.  The  sum  of  the  ages  of  A,  B,  and  C  is  108  years.  A  is  twice 
as  old  as  B,  and  twice  C's  age  is  equal  to  A's  and  B's  together. 
Find  their  ages. 

10.  After  paying  2  %  taxes  on  my  income  I  have  $  1078  left. 
What  is  my  income  ? 

11.  I  pay  33  J  %  duty  on  the  cost  price  of  a  horse.  I  keep  him 
2  months  at  an  expense  of  $16,  and  I  then  sell  him  for  $200,  mak- 
ing 20  %  profit  on  the  cost  price.     What  did  the  horse  cost  ? 

12.  In  a  certain  school  f  of  the  pupils  are  in  the  first  form,  J  in 
the  second,  ^^  in  the  third,  and  14  in  the  fourth.  How  many 
pupils  are  in  the  school  ? 

13.  A  market-woman  sells  to  A  half  an  egg  more  than  half  she 
has,  to  B  half  an  egg  more  than  half  she  has  left,  and  10  eggs  to 
C,  and  she  then  has  6  eggs  left.     How  many  had  she  at  first  ? 


CHAPTER  11. 

The  Four  Elementary  Operations, 
addition  and  subtraction. 

24.  The  addition  of  a  and  h  is  denoted  by  a+&,  where 
a  and  h  stand  for  any  quantities  whatever. 

If,  however,  a  =  5  and  6=  —3,  the  expression  becomes 
5  —  3,  and  we  have  a  case  of  subtraction. 

Thus,  symbolically,  addition  and  subtraction  are  one 
and  the  same ;  for  in  the  expression  a  +  6  we  cannot 
know  whether  an  addition  or  a  subtraction  is  to  be  per- 
formed, until  we  know  something  about  the  quantities  for 
which  the  letters  stand. 

Moreover,  any  subtraction  may  be  put  into  the  form 
of  an  addition,  and  vice  versa;  for  a  —  h  is  the  same  as 
a  +  (  —  ?>),  and  a  +  6  is  the  same  as  a  —  {—h). 

Thus  the  subtraction  of  one  expression  from  another 
may  be  expressed  as  an  addition,  by  changing  all  the 
signs  of  the  subtrahend.  Hence  the  rule  for  Algebraic 
Subtraction : 

Change  the  signs  of  the  subtrahend^  and  then  perform 
addition. 

Ex.  To  subtract  3a— 2  6  +  3  from  6a+36— 4  is  the  same  as  to 
add  — 3a4-2  5— 3  to  6a+3  6— 4;  and  the  result  is  3  a+ 5  &  — 7. 

25.  Since  a  series  of  terms  connected  by  +  and  —  signs 
may  be  wptteix  as  one  connected  by  +  signs  only,  such 

17 


18     THE  FOUR  ELEMENTARY  OPERATIONS. 

a  series  is  called  an  Algebraic  Sum.  Thus  5  —  4  +  3  —  1 
has  3  as  its  algebraic  sum,  and  may  be  written 

5  +  (-4)  +  3  +  (-l). 

We  are  not  justified  in  speaking  of  this  as  an  Arith- 
metic  Sum,  for  —  4  and  —  1  have  no  meaning  in  pure 
arithmetic,  and  if  we  put  it  into  the  form  S  —  {-{-5), 
it  becomes  an  arithmetic  difference. 

26.  Symmetry.  When  the  interchanging  of  two  let- 
ters of  an  expression  leaves  the  expression  unchanged, 
except  as  to  the  order  of  the  letters  in  a  term,  or  the 
order  of  the  terms  in  the  expression,  the  expression  is 
symmetrical  in  the  two  letters. 

Thus,  by  interchanging  a  and  b  in 

ab  —  ac -\-  ad  —  be  -\- bd  —  cdy 

we  get  ba  —  be -\- bd  —  ac -\-  ad  —  cd, 

an  expression  the  same  as  the  former,  except  as  to  the 
order  of  the  terms  and  of  the  letters  in  some  of  the  terms. 

Hence  the  expression  is  symmetrical  in  a  and  b. 

It  is  readily  seen  that  the  expression  is  not  symmet- 
rical in  c  and  d. 

An  expression  which  is  symmetrical  in  every  pair  of 
two  letters  is  symmetrical  in  all  the  letters. 

Thus  ab  -\-bc-{-  ca  and  abc  +  abd  +  acd  +  bed  are  each 
symmetrical  in  all  the  letters  which  they  contain. 

Some  special  kinds  of  symmetry  will  be  considered  in 
a  more  advanced  stage  of  the  work. 

27.  When  we  know  the  letters  which  enter  into  an 
expression  symmetrical  in  them  all,  and  we  are  given  a 
type-term,  we  can  write  the  full  expression  by  building 


ADDITION    AKD   SUBTRACTIOK.  19 

up  the  form  of  the  type  in  every  possible  way  from 
the  given  letters,  and  taking  the  algebraic  sum  of  all 
the  terms  so  produced. 

Thus  from  the  letters  a,  b,  c: 

i.    with  type  ab^-  we  have  — 

ii.  with  type  a  {he  —  a)  : 

a  {he  —  a)  +  h  (ca  —  h)+  c  (ah  —  c). 
iii.   with  type  ahc^i 

ahc^  +  hca'^  +  cah'^. 
iv.   with  type  (h  —  c)  (a^  _  5c)  : 

(6  -  c)(a2  _  hc)-{-(ic  -  a)(h-^  -  ca)  +  {a  -  h){c^  -  ah). 

V.   with  a,  6,  c,  and  d,  and  type  ah'^ : 

a&2  4.  5^2  _}_  ^(.2  _|_  ca2  _|.  (;r(^2  _|_  ^^2  ^  5^2  +  c62  +  6d^ 

+  d62  4-  cd^  4-  (^c2. 

It  will  be  noticed  that  in  examples  ii.  and  iii.  each  term 
is  formed  from  the  preceding  one  by  changing  a  to  h, 
b  to  c,  and  c  to  a.     This  is  called  a  cyclic  or  cir-         „ 
cular  substitution ;  for  if  we  write  the  letters    /         \ 
in  a  circle,  as  in  the  margin,  we  pass  from  one  (  , 

term  to  the  next  by  commencing  with  a  letter   '''^^^ — ^* 
one   step  further  around  the  circle  until  the  whole  is 
completed. 

Many  substitutions,  where  three  letters  are  concerned, 
are  of  this  character,  the  distinctive  feature  being  that 
we  do  not  interchange  any  two  letters  without,  at  the 
same  time,  interchanging  every  two  in  circular  order. 

A  cyclic  change  with  4  letters  and  type  ab  gives 
ab -\- be -{- cd -\-  da.  This  lacks  the  terms  ac  and  bd  to 
make  it  completely  symmetrical. 


20  THE  FOUR   ELEMEKTARY  OI^ERATIONS. 

In  examples  i.  and  v.  a  circular  change  is  not  suffi- 
cient ;  for  from  the  type  ab^  we  must  have  a  term  &a^, 
which  is  not  given  by  a  mere  circular  substitution.  In 
other  words,  we  must  interchange  two  letters  without 
affecting  the  third. 

A  little  care  and  observation  are  all  that  are  required 
in  writing  out  such  expressions  from  a  given  type. 

28.  The  symbol  2  (sigma),  amongst  other  uses,  is 
conveniently  employed  to  denote  expressions  consisting 
of  algebraic  sums,  written  from  a  type. 

These  are  symmetrical  in  all  the  letters  employed,  and 
when  written  out  are  frequently  of  inconvenient  length. 

The  notation  ^a^b,  with  three  letters  involved,  stands 
for  a^b  4-  b^a  -f  b^c  +  c^b  +  c^a  +  a^c. 

With  four  letters  involved  it  stands  for  v.  of  the  pre- 
ceding article. 

^{b  —  c  '  a^  —  be)  stands  for  iv.  of  the  preceding  article. 

As  employed  hereafter,  3  letters  will  be  understood  un- 
less a  different  number  is  indicated,  or  in  cases  where 
misunderstanding  is  not  possible.  ^4  will  serve  to  indi- 
cate  4  letters,  and  generally  2n  to  indicate  n  letters. 

This  is  known  as  the  Sigma  Notation. 


EXERCISE 

II. 

a. 

1. 

With  3  letters  write  in  full  — 

1. 

S(a2_5). 

iv. 

(Sa)2 

ii. 

2  («  +  6  - 

C)2. 

V. 

{2  (a 

-b)f. 

ill. 

c 

vi. 

2a  X 

^ab. 

vii.    (2a)-2  +  2  (a  +  &  -  c)2  -  4  2a2  +  2  Zab. 
yiii.   x^  —  x22a  +  x^ab  —  abc. 


MULTIPLICATIOK.  21 

2.    With  4  letters  write  in  full  — 

i.    Sa6.  iv.  (2a)2. 

u.    Sa&2c.  V.  2  (a  -  &)  (c  -  (T). 

iii.   Sa(5-c).  vi.  ^{aP' -h)(d^  -  d). 

MULTIPLICATION. 

29.  The  multiplication  of  one  expression  by  another 
may  be  effected  by  a  series  of  distributions,  and  when  the 
expressions  are  not  too  complex  this  is  usually  the  best 
method. 

Thus   (a  -h  &)  (a  —  &  +  c)  =  a(a  —  &  +  c)  +  &(a  —  6  +  c) 
=  a^  —  ah -{- ac -{- ah  —  h^  -\- be  =  a^  —  h'^-{-ac-{-  he. 

By  remembering  and  applying  a  few  elementary  pro- 
duct forms,  the  operation  may  often  be  much  curtailed. 
As  convenient  fundamental  forms  we  may  take  the  fol- 
lowing, although  any  simple  form  that  can  be  remembered 
may  be  equally  useful : 

(1)  {a  +  by  =  a'-^h'  +  2ab, 

(2)  (a-hy  =  a'-\-h'-2ah. 

(3)  (a-h){a  +  h)  =  a:'-h\ 

(4)  3  (a  +  &)  (&  +  c)  (c  +  a)  =  (a  +  5  +  c)^-  a^-  6^-  c^ 

(5)  (a  +  h  +  c)  (a2  ^h^  j^c^ -ah -be-  ca) 

=  a^^b^-{-c^-3abc, 
or  Sa  (Sa^  —  5a&)  =  ^a^  —  S  abc. 

The  mark  .-.  is  a  verbal  symbol  for  'therefore'  or 


hence.^ 


22     THE  FOUR  ELEMENTARY  OPERATIONS. 

Ex.  1.    {a  -\-  b  -  c  +  d)(a  +  b  +  c  -  d) 


^(a  -^  b  -  c  -  d)(a  -\-  b  +  c  -  d) 

=  (a  +  6)2-(c-(^)2 

=  a2  +  52  -  c2  -  (Z^  +  2  a6  +  2  c(^. 

Ex.  2.     (a  +  &  +  c  +  dy  =  (a  +  6)2+  2(a  +  6) (c  +  c?)  +  (c  +  (?)2 
=  a2  ^  52  ^  c2  ^  ^2  _|_  2(a&  +  ac  4-  a(^  +  &c  +  6(^  +  cd). 

Ex.  3.   To  distribute 
s(s— a)(s— 5)  +  s(s— 5)(s— c)+-s(s— c)(s— a)  — (s— a)(s— ?))(s— c), 
where  2  s  =  a  +  5  +  c.    s,  being  symmetrical  in  a,  6,  and  c,  is  not 
altered  by  a  circular  substitution  of  these  letters. 

But  this  substitution  brings  s(s  —  a)(s  —  b)  to  s(s  —  b)(s  —  c), 
and  s(s  —  6)(s  —  c)  to  s(s  —  c)(s  —  a). 

Hence  having  the  expansion  of  s(s  —  a)(s  —  b),  the  expansions 
of  the  two  following  terms  may  be  immediately  written  down  by 
a  cyclic  interchange  of  letters. 

Now  s(5  —  a)(s  —  b)  =  s^  —  s^(a  -j-  b) -\-  s  -  ab. 

.-.  s(s  -b)(s-  c)  =  s^-  s%b  +  c)+  S'bc, 

and  s(s  —  c)  (s  —  «)  =  ^  —  ^""^(^  +  a)  +  s  •  ca. 

The  algebraic  sum  is 

3s3_2s2(^a  +  5_l_c)+s(a&  +  6c+ca), 
and  as  2s=:a  +  5  +  c,  this  becomes 

—  s^  +  s(a6  +  bc-j-  ca) A 

Again  the  expansion  of  (s  —  a)(s  —  b)  (s  —  c)  is 

^  —  s2(a  +  6  +  c)  +  s(ab  +  6c  +  ca)  —  abc, 
or  —  s^  +  s(a6  +  &c  +  ca)  —  abc. 

And  subtracting  this  from  A  gives  a6c  as  the  final  result. 

This  example  furnishes  a  good  illustration  of  the 
remark  in  Art.  20,  for  the  long  series  of  operations  sym- 
bolized in  the  statement  of  the  exercise  is  just  equiva- 
lent in  its  totality  to  the  two  multiplications  symbolized 
in  the  final  result. 


MULTIPLICATION.  28 

Ex.  4.  To  prove  the  identity 

8(2a)8  -  S(a  +  by  =  S(2a  +  b  +  c)(2  6  +  c  +  a)(2c  +  a  +  b). 

Either  of  two  methods  may  be  adopted —  (1)  to  transform  one 
member  to  the  other  by  the  rules  of  operation  ;  or  (2)  to  transform 
each  to  the  same  third  expression  by  distribution.  We  shall  adopt 
the  first  way. 

8(2a)3^(S2a)3=  (a  +  b  +  b  +  c  +  c -\-  a^. 

Now  put  a  +  b  =  py  b  +  c  =  q,  c  -\-  a  =  r,  and  the  identity 
reduces  to 

(p  +  q  +  ry  - P^  -  q^  -r^  =  3(j)  +  r)  (r  +  g) (g  +  i?), 
which  is  true  by  Art.  29,  (4). 

30.  Expansion  of  Symmetrical  Homogeneous  Expres- 
sions. 

This  form  of  expression  is  of  frequent  occurrence,  and 
its  properties  of  symmetricality  and  homogeneity  enable 
us  to  expand  it  with  some  facility. 

Ex.  1.  To  expand  (a -{- b -\- cy  -  (a  +  by  -  (6  +  cy  -  (c  +  ay. 

Being  homogeneous  and  of  3  dimensions,  the  type  terms  in  its 
expansion  can  only  be  a^,  a^b,  and  abc.  Taking  the  type  a^,  we 
see  that  its  coefficient  is  —  1.  Taking  the  type  a'^b,  its  coeffi- 
cient is  readily  found  to  be  zero.  And  the  coefficient  of  the  type 
abc  is  6. 

.-.  The  expansion  is  6  abc  —  a^  —  b^  —  (^. 

Ex.  2.  To  expand  (Sa)^  +  («  +  5  _  c) (6  +  c  -  a) (c  +  a  -  6). 

The  expansion  being  homogeneous  of  3  dimension^  and  sym- 
metrical, must  be  of  the  form 

m2a3  4-  nSa^ft  +  p  •  abc. 

The  coefficient  of  a^    is  1  —  1  or  0 ;    .*.  m  =  0. 

The  coefficient  of  a-5  is  3  +  1  or  4  ;    .*.  n  =  4. 

The  coefficient  of  abc  is  6  —  2  or  4  ;    .-.  p  =  4, 
and  the  expansion  is  4  'Za'^b  +  4  abc. 


24  THE  FOUR   ELEMENTARY   OPERATIONS. 

EXERCISE  II.  b. 

1.  Write  out  the  type  terms  in  the  following  symmetrical  and 
homogeneous  expressions  — 

i.   (a-{-b  +  c  +  ay.  iii.   (a  +  6  4-  c)*. 

ii.   (a  +  by.  iv.    (a+b  +  c  +  dy. 

V.  (a  +  b  +  c  +  ay, 

2.  Show  that 

(a  -j-  b  —  c)(b  -{-  c  —  d)(c  +  a  —  b) 

=  ab(a  +  b)+  bc(b  +  c)  +  ca(c  +  a)  -  a^  -  6^  -  c^  -  2  ahc, 

3.  Sa- 2a6-(a  + &)(& -f-c)(c  +  a)=a6c. 

4.  Show  that  S{(a  -  5)  (2  &  -  c)}  =  (Sa)2  -  3  Sa^. 

5.  Show  that 

(a  +  6  +  c)  (a  +  &  -  c)  (6  +  c  -  a)  (c  +  a  -  6) 

=  2Srt252  _  Sa4  =  a2(2&2_  a2)^  52(202  -  b'^)^d^{^aP'  -  d^) 

=  4  52c2_(62  +  c2-a2)2. 

6.  Expand  — 

i.  S(a  + &)(«-&). 
ii.  (Sa)*  -  S(a  +  &)* -f  Sa*. 
iii.  Sa3(6_c)-Sa.Sa2(6_c). 
iv.  a(s  —  6) (s  —  c)  +  6(s  —  c) (s  —  a)  +  c(s  —  a) (s  —  5) 

+  2(s  —  a) (s  —  5) (s  —  c),  where  2s=:a  +  &  +  c. 

7.    If  2  s  =  a  +  &  +  c,   show  that  the  three  following  expres- 
sions are  identical  in  value  — 

s(s  —  d)(b  -\-  c)  +  «  (s  —  5)  (s  —  c)  —  2  bcs, 

s  (s  —  6  )  (c  +  a)  +  6  (s  —  c)  (s  —  a)  —  2  cas, 

s  (s  —  c)  (a  +  6)  +  c  (s  —  a)  (s  —  6)  —  2  a6s. 


MULTIPLICATION.  25 

8.  Show  that  (x  —  b)(x  —  c)  (b  —  c)  +  (x  —  c)  (x  —  a)(c  —  a) 
-{■{x-a){x  -h){a-h)  +  (a  -  6)(6  -  c)(c  -  a)=0. 

9.  Show  that  (2a)2  =  2^2  _|_  2  2a&,  with  any  number  of  letters. 

10.  Multiply  2a2  +  2  2a6  by  2a,    3  letters. 

11.  Multiply  2a2  +  2a&     by  2a6,  3  letters. 

12.  Multiply  2a2  4.  2  2a6  by  2a,    4  letters. 
This  gives  (a  +  &  +  c  -f  ^)^. 

Prove  the  following  theorems  in  numbers  — 

13.  The  difference  between  the  square  of  the  sum  and  the 
square  of  the  difference  of  two  numbers  is  the  product  of  twice 
the  numbers. 

14.  The  sum  of  the  squares  of  the  sum  and  of  the  difference 
of  two  numbers  is  one-half  the  sum  of  the  squares  of  twice  the 
numbers. 

15.  If  two  numbers  be  each  the  sum  of  two  squares,  their  pro- 
duct is  the  sum  of  two  squares. 

16.  If  two  numbers  be  each  the  difference  between  two  squares, 
their  product  is  the  difference  between  two  squares. 

17.  If  the  sum  of  two  numbers  is  1,  their  product  is  equal  to 
the  difference  between  the  sum  of  their  squares  and  the  sum  of 
their  cubes. 

18.  If  the  product  of  two  numbers  is  1,  the  square  of  their  sum 
exceeds  the  sum  of  their  squares  by  2. 

31.  The  distribution  of  (a?  +  a)  {x  +  6)  (a?  +  c)  ...,  the 
product  of  a  number  of  binomial  factors  with  one  letter 
the  same  in  each  factor,  is  very  important. 

The  dominant  letter,  x,  is  taken  as  the  variable,  and 
the  expansion  is  arranged  according  to  the  powers  of  x. 

With  three  factors  we  readily  see  that  taking  the  x 
from  every  factor  gives  ^)  taking  it  from  every  factor 


26     THE  FOUR  ELEMENTARY  OPERATIONS. 

but  one,  and  taking  the  other  letter  from  that  one  gives 
Qc^  {a -\- b  -{- c)  ',  taking  x  from  one  factor  and  the  other 
letter   from   the  other  two  gives   a;  {ab  +  bc-{-  ca)  ;   and 
lastly,  taking  the  second  letters  only  gives  abc. 
Thus  the  expansion  is 

a^  +  aj^  (a  -f  6  +  c)  +  X  (ab  +  ^c  +  ca)  -f  abc, 

or         a^  -{■  a^^a  +  x^ab  +  abc. 

Similarly,  with  4  factors  it  is  readily  seen  that  the 
expansion  is 

aj*  +  a^%a  +  a^2a6  -f-  x^abc  +  abed. 

In  a  similar  manner  it  is  shown  that  with  n  factors 
the  expansion  is 

^n  _|_  ^n-i^a  4-  x"  22a6  +  a;"-^2a6c  H \-abC'". 

It  will  be  noticed  in  every  case  that  the  last  term  is 
the  continued  product  of  all  the  letters  except  the  variable. 
This  is  important. 

If  the  signs  are  negative  in  all  the  factors,  as  (x  —  a), 
etc.,  then  Sa,  labc,  etc.,  involving  an  odd  number  of 
letters  in  each  term,  will  be  negative,  and  2a6,  etc.,  in- 
volving an  even  number,  will  be  positive. 

Thus,  {x  —  a)  (x  —  b).(x  —  c)  (x  —  d)  •  •  •  to  n  factors 
=  aj~  —  a;"-^2a  +  x^'-^^ab  —  H 

Ex.  1.    (x  +  a)(x  +  b)  (x  +  c) 

=  x^  +  ic2(a  4-  2)  -f  c)  +  x(ab  -\-bc -{-  ca)  +  abc ; 
and  making  c  =zh  =  a  gives 

(x  +  ay=x^  +  Sx'^a  +  Sxa'^  +  a^ 

Ex.2.    (x  +  a)(ix  +  b)(ix-\-c)(ix  +  d) 

=  X*  +  x^Xa  +  x'^Zab  +  x^abc  +  abed. 


MULTIPLICATION.  27 

Now  2a  contains  4  terms,  2a6  contains  6,  2a&c  contains  4,  and 
a5cd  is  one  term. 

Therefore  making  d  =  c  =  h  =  a  gives 

(x  +  a)*  =  X*  +  4  ic%  +  6  x2a2  _|-  4  ^cq^s  _}.  gi^ 

Ex.  3.   Similarly,  we  find 

{x  -\-ay  =  x-5  +  5  x*a  +  10  x^a^  +  10  x^a^^  +  5  xa^  +  a'\ 

The  coefficients  of  the  several  terms  in  these  and 
higher  powers  are  exhibited  in  the  following  table,  which 
may  be  extended  at  pleasure  — 

The   coefficients   form   the   diagonals,  up  to  the  8th 
power. 
1     ^^.-'1      ^^--'i     ^^.-'1       ^.-'1       ^^^'1       ^^^'1        ^^^'1       ^^^'1 

1"'''  ^^'-3^'"  ^-^^'''  '     --''°''''     ^-■'5'"''    ^-21''''    ^.28'"' 
r"'      ^-'^^'^    .-'lO''"    ^.^O'"'  ^^.35-'""  ^,'56"""' 
1"''   ^^--5""   ^--15''''"    ^^35''"    ^.-TO'""' 
1''"'      .6''"''    .21'''"''    ^,56""' 

I''""'  ^,-8'"'" 

Ex.  4.   (a;  +  a  -  6)(x  +  5  -  c)(a;  +  c  -  a)  =  ic3  +  a^s^^^  _  5) 

+  x2{(a  -  5)(6  -  c)}  +  (a  -  &)(&  -  c)(c  -  a). 
Now      2(a  -  5)=  0,  2{(a  -  &)(6  -c)}  =  2a6  -  2a2, 
and  (a  -  6) (6  -  c) (c  -  a)  =  2a5(6  -  a), 

and  the  expansion  is 

x^  +  a:(2a6  -  2a2)  +  2a6(&  -  a) 

EXERCISE  II.  c. 

1.  Expand  (a;  _  l)(x  -  2)(x  -  3)(x  -  4). 

2.  Expand  (x  -  l)(a^  +  2)(x  -  3)(ic  +  4). 

3.  Expand  (4a:  +  l)(3x  +  2)(2x  +  3)(a;  +  4). 


28  THE   FOUR   ELEMENTARY    OPERATIONS. 

4.  Expand  (x+a  +  &  —  c)(x  +  5  +  c  —  a)(x  +  c  +  a  —  6). 

5.  Expand  (a  -\-  b  +  c)^ 

Write  it  {«+(&  +  c)}^  and  pick  out  the  coefficients  of  the  type 
terms. 

6.  Expand  (a  +  6  +  c  +  d)*. 

Write  it  {(a  +  6)  +  (c  +  d)}*  and  pick  out  the  coefficients  of  the 
type  terms. 

7.  Write  in  S  notation  the  expansion  of  (a  -f  6  +  c  +  d)^. 

32.  Function.  An  expression  such  as  a^  +  ab  changes 
value  when  a*  changes  value  or  when  b  changes  value. 
It  is  accordingly  called  a  function  of  a  and  b. 

When  we  wish  to  consider  a  alone  as  a  variable,  and 
regard  b  as  being  constant,  we  speak  of  the  expression 
as  a  function  of  a,  and  we  symbolize  it  as  fa  or  /(a), 
where /is  di  functional  symbol. 

This  symbol  merely  denotes  that  a  enters  into  an 
expression  as  a  variable,  without  any  regard  to  the  other 
letters  in  the  expression,  and /(a;)  stands  for  any  expres- 
sion in  which  x  enters  as  a  variable. 

In  general  algebra  the  form  of  /  is  given,  i.e.  we  are 
given  an  expression  of  the  form  required. 

Thus,  if /(a)  stands  for  a^ -{- 2  ab -\- c,  then /(a;)  stands 
for  x^  -\-  2  xb  +  c,  where  x  is  substituted  for  a  in  the  type- 
form.    Similarly,  f{a  —  x)  =  {a  —  x)^ -\-  2  (a  —  x)  b  -\-  c,  etc. 

Ex.1.  If /(a)  =  aH2a  +  l,  /(a-l)  =  (a-l)2+2(a-l)  +  l=a-2. 

EX.2.  If  /(.)=^,  /(r3^)=(r^J/(i+r^)=x. 

EXERCISE  II.  d. 

1.  If  f(x)=x^  +  3x  -  10,  find  /(3),  also  /(-  3). 

2.  If  fix)  =  x2  —  5x  4-  6  and  y=Z  —  x^  find  /(?/)  in  terms  of  x. 


MULTIPLICATION.  29 

3.  If  f{a)  =  1  -  a,  show  that  /{/(«)}  =  a, 

4.  If  f{x)  =  a-x,  show  that  p(x)  =  f{x), 

P  stands  for  /{/(/)}. 

5.  If  f{x)  =  x^  +  X  +  1  find  f(x  -  1). 

6.  If  fix)  =  1  +  X  +  —  +      ^^      + — +  etc.,  find  the 

*^^  ^  1.2       1.2.3      1.2.3.4 

numerical  value  of  /(I)  to  5  decimal  places. 

7.  If  /(x)  =  a;4  _  3x3  +  2x2  +  3a:  -  3,  find  f(x  +  1). 

33.  An  expression  such  as 

or  l-2aj-3aj2  +  2a.^  +  aJ*, 

in  which  the  exponents  of  the  variable,  x,  are  all  positive 
integers,  is  called  a  positive  integral  function  of  x,  or 
simply  an  integral  function  of  x. 

The  first  of  these  is  written  in  descending  powers  of 
the  variable,  and  the  second  in  ascending  powers. 

The  coefficients  may  be  numerical  or  literal. 

The  function  is  complete  when  all  the  powers  of  the 
variable  in  consecutive  order  are  represented;  and  any 
integral  function  may  be  made  complete  in  form  by  writ- 
ing zero  coefficients  to  the  missing  terms.     Thus, 

aj5  ^  Oic*  +  2ar^  +  Oic^ -  3a;  +  1 

is  complete  in  form. 

34.  In  multiplying  together  two  integral  functions  of 
the  same  variable,  it  is  advantageous  to  operate  upon  the 
coefficients  alone,  as  the  proper  powers  of  the  variable 
are  readily  supplied  to  the  result. 

The  functions,  if  not  complete,  should  be  made  com- 


30 


THE  FOUR  ELEMENTARY  OPERATIONS. 


plete  in  form,  and  there  is  some  advantage  in  writing 
them  in  ascending  order  of  the  variable. 


Ex.  1.  To  multiply  a -\- bx  +  cx'^  +  dx^  ^J  P  +  qx -{-  rx^. 

h        +c  +d 

■q        +r 


Coefficients 


fa    +1 
\p    +( 


Product    withf^«  +  ^^l^+^^^ 
variable  supplied  j  ^  ^ 


4-  ra 


X2+J 


+  q(' 
+  r6 


+  re  I 


By  observing  in  this  typical  case,  how  the  coefficients 
in  the  product  are  made  up,  i.e.  by  a  sort  of  cross-multi- 
plication, as  pb  +  qa,  pc-{-qb  +  ra,  etc.,  we  can  perform 
such  multiplications  with  numerical  coefficients  with 
considerable  facility. 

Ex.  2.  To  multiply  1  -  2x  i- Sx'^ -{- x^  hy  2-x-{-2x^. 

1-2+3      +1 Multiplicand. 

2-1+2  Multiplier. 

2  -  6x  +  10x2  -  5x3  +  5x*  +  2x5    .  Product. 

Ex.  3.  To  multiply  x2  +  Jx  +  J  by  x2  -  Jx  -  |. 
Arranging  in  ascending  powers  — 

-i-iic-ix2  +  0x3  +  x* 

Ex.  4.  To  expand  (i  +  x  -  2  x2  +  3  x^y. 

1st  operation  P  +  ^  ~  ^  +  ^ 

ll  +  l_2  +  3 

2d  operation  P  +  2-3  +  2  +  10-12  +  9   1st  product. 

11  +  1-2  +  3 

1  +  3-3-2  +  24-15-17  +  63-54  +  27 
Result,  l+3x-3x2-2xH24x*-15x5-17xH63x7-64x8+27x«. 


MULTIPLICATION.  31 

Ex.  5.  To  multiply  x'^  -  2x^  -\- x  +  1  by  x'^  +  1. 
Ordering  in  descending  powers  — 

1+0+0-2+0+1+1 

1  +  0-f  1 

1+0+1-2+0-1+ 1+1+1 
Result,  x^  +  x^  -  2x^  -  x^  -{-  x^  +  X  -{- 1. 

35.  A  circulating  decimal,  or  the  arithmetical  approxi- 
mation to  the  value  of  any  incommensurable,  is  an  example 
of  a  series  of  arithmetical  figures  which  is  non-termina- 
ting. Similarly,  in  algebra  we  may  have  a  series  of 
terms,  arranged  in  ascending  powers  of  the  variable, 
such  that  the  series  has  no  last  term.  Such  a  series  is 
called  an  infinite  series,  and  is  indicated  by  writing  a  few 
terms  at  the  beginning  with  three  points,  •••,  with  or 
without  ad  inf. ;  as 

a-\-bx-{-  cx^  +  dar^  +  • .  •  ad  inf. 

As  we  cannot  write  all  the  terms  of  an  infinite  series, 
we  cannot,  in  general,  write  all  the  terms  of  any  multiple 
of  it.  In  some  cases,  however,  certain  multiples  may 
become  finite  by  the  vanishing  of  all  the  terms  after  the 
first  few. 

We  have  the  arithmetical  analogue  in  a  repeating  or 
circulating  decimal,  such  as  1.2333---,  which  gives  a 
finite  product,  3  •  7,  when  multiplied  by  3,  but  another 
infinite  series  when  multiplied  by  4  or  5. 

To  multiply  two  infinite  series  together,  we  take  the 
same  number  of  terms  in  both  multiplicand  and  multi- 
plier, and  retain  that  number  of  terms  in  the  product. 


32 


THE  FOUR  ELEMENTARY  OPERATIONS. 


by 


Ex.  1.   To  multiply 


l-\-X  +  x'^4-X^  +  X^  + 
1-x  +  x^  -X^-\-X^  - 

^  ll  _1  +  1_1  +  1 


Product,  1  +  »2  +  ic*  + 


1+0+1+0+1 
,  an  infinite  series. 


Ex.  2.   To  multiply  1-^2  +  2  x^  -  3  x*  + 

by  1  +  2  X  +  ic2. 

14-2-3... 


Operation  P  +  ^      ^ 

^  ll  +  2+  1 


1+2  +  0  +  0+0... 
Product,  1  +  2  x,  a  finite  result  as  far  as  the  series  extends. 

36.   To  square  the  series  a -\-  bx -\-  cx^  -\-  dx^  -{-  ex^  -\- ' 
Operation    {"  +^    +'     ^'^     +'  +  "" 


Coefficients  < 


4-6    +c      4-^      +e  + 


a?  -\-2ab-\-2ac 


+  ( 


-\-2ad 

-\-2ae 

-{-2hc 

+  2hd 

+  0^ 

+ 


The  operation  is  simply  multiplication,  but  owing  to 
the  identity  of  the  multiplier  and  the  multiplicand,  the 
coefficients  in  the  result  consist  of  double  products  and 
squares  ;  and  we  notice  that  a  square  appears  as  the  first 
coefficient,  and  then  in  every  alternate  one. 

By  observing  how  the  coefficients  are  made  up,  we 
may  write  the  square  of  a  series  with  great  ease. 

Ex.  1.  To  square  1  +  2ic  +  Sic^  +  4x3  H to  the  term  con- 
taining x^. 

1+2+3     +    4 


1  +  4  +  6   1+    8|  +  ... 
+  22  I  +  12  I 
.-.  1  +  4x  +  10x2  +  20x^  +  •••  is  the  required  square. 


MULTIPLICATION.  83 


Ex.  2.  To  square  1  +  hx  -  Ix'^  +  j\x^  ■ 
1  +  i-J    +tV 


/.  Square  =  1  +  ». 

EXERCISE  II.  e. 

1.  Multiply        1 +  2ic  +  3x2  4-4x3  +  6x* 
by  l-2x  +  3x2-4x3  +  6x*. 

2.  Multiply        l-x  +  ix'^-7x"+  19x*  -  40x5  +  ... 
by  l  +  ic-3x2, 

to  the  term  containing  x^. 

3.  Multiply        -  1  -  X  +  3x2  -  2x8  -  X*  4-  3x5  -  2x6 

by  1  +  X  +  x2, 

to  the  term  containing  x^. 

4.  In  the  complex  series 

1  +  m(ax  +  6x2  ^  cx3  +  ...)  -|-  n  (ax  +  hx^  +  cx^  +  ...)2 

+  p{ax  +  6x2  +  0x3+  '••y-\-' 
find  the  coefficient  of  x'. 

5.  Find  the  coefficient  of  x^y*  in  the  product  of 

(x  +  ax3  +  6x5  +  ...)  by  /l  _^- ^^-Lly4...y 

6.  Square  the  series,        1  +  ^  "  |(f )'+  |(|)'- 

7.  Multiply        1  +  x(l  -  2  x)  +  x2  (1  -  2  x)2  +  ... 
by  1  -  X  +  2  x2, 

to  the  term  containing  x^. 
Let  2/  =  X  —  2  x2. 


34     THE  FOUR  ELEMENTARY  OPERATIONS. 

8.  Find  the  square  of    x — ^ ^ ^ ?-^ 

2x      1-2  2-^x^      1.2.8   23^5 

The  law  of  formation  of  the  terms  is  evident,  and  any  required 
number  of  terms  may  be  written  down. 

9.  Show  that 

(1  -  2x  +  3x2  -  4a;3  +  ...)(^1  +  2a;  + 3x2  +  4x3  +  •••) 

=  (H-X2  +  X*  +  -)^- 

10.  Find  the  coefficient  of  x"  in  the  product 

(1  +  CiX  +  C2X2  +  C3X3  +  ...)(X~  +  CjX"-!  +  C2X»-2  +  CgX^-S  +  ...). 

11.  Find  the  coefficient  of  x"-^  in  10. 

12.  Find  the  coefficient  of  linear  x  in 

l  +  ^^  +  ^C^-l)^2_^^(x-l)(x-2)     3      ^^^ 
1.2  1.2.3 

13.  If    y  =  ax-[-  bx^  +  cx^  H and   x  =  Ay  +  By'^  +  Cy^  -\ 

find  A  and  B  in  terms  of  a  and  6,  on  the  condition  that  the  coeffi- 
cient of  each  power  of  x,  in  the  result  of  substituting  for  ?/,  is  zero. 

14.  If/(x)^6|l  +  l.J-+ll^.-L  +  Ll3:i._L+...l 

*^^^         Ix      1    3x*      1.2    5x'      1.2.3    7x10^       / 

calculate  /(2)  to  four  decimal  places. 

15.  If  /(x)  =  x  —  Jx^  +  ^x^  —  ix^+  ...,  calculate,  to  four  deci- 
mal places,  the  value  of  8{/(i)  +  /(J)}  +  4/(|). 

In  order  to  obtain  4  decimals  exact,  the  calculation  should  be 
carried  to  at  least  5  places. 

16.  In  any  multiplication,  write  the  terms  of  the  multiplier  in 
an  inverted  order,  and  the  partial  products  are  not  formed  by  a 
cross-multiplication.    How  are  they  formed  ? 


DIVISION.  85 


DIVISION. 

37.  Division,  in  algebra  as  in  arithmetic,  is  indicated 

in  several  ways.     Thus,  a-i-b,  -,  and  a/b  all  mean  that 
a  is  to  be  divided  by  b. 

In  any  case,  a  is  the  dividend  and  b  the  divisor,  where 
a  and  b  stand  for  any  numerical  quantities  or  algebraic 
expressions. 

We  define  the  operation  indicated  by  -  as  the  inverse 

b 

of  multiplication,  such  that  -  xb  =  a. 

b 

Denoting  -  by  g,  we  have  a  =  bq,  where  q  is  the  quo- 
b 

tient.     Hence  the  theorem : 

The  dividend  is  the  product  of  the  divisor  and  the 
quotient,  and  the  quotient  and  the  divisor  are  reciprocals 
in  the  sense  that  if  either  be  made  the  divisor  the  other 
is  the  quotient. 

Division  thus  consists  in  separating  the  dividend  into 
two  factors,  one  of  which  is  the  divisor ;  and  any  process 
which  accomplishes  this  effects  the  division. 

38.  Index  law  in  division.  Assume  —  =  a^,  and  multi- 
ply both  members  by  a^ 

Then  a'"  =  a"a^  =  a"+^,  by  the  index  law. 

Therefore  m  =  n  -\-  p,  ov  p  =  7n  —  n. 

Whence  —  =  a"*"" ; 

a" 

and  this  must  hold  for  all  integral  values  of  m  and  n. 


36     THE  FOUR  ELEMENTARY  OPERATIONS. 

Hence  the  quotient  from  dividing  any  integral  power 
by  another  integral  power  of  the  same  root  is  that  power 
of  the  root  whose  index  is  found  by  subtracting  the 
index  of  the  divisor  from  that  of  the  dividend. 

Cor.  1.  If  m  =  71,         —  =  1  =  a"-~  =a*^. 
a" 

Hence  the  zero  power  of  any  finite  quantity  is  to  be 
interpreted  as  meaning  +  1. 

Cor,  2.  Making  m  =  0,     —  =  ^  =  a<^  *»  =  a-\ 
a""     a"" 

Hence  a  negative  exponent  is  to  be  interpreted  as  the 
reciprocal  of  the  same  root  with  the  corresponding  posi- 
tive exponent. 

Thus,  ?  =  a6-i;  1 +-  + ^=1 +  ^^ +  ^  ^  etc. 

39.  The  most  important  cases  of  division,  where  any 
special  process  is  required,  are  those  involving  a  variable 
in  an  integral  function. 

Let  aoi?  +  6a?  4-  c  be  a  divisor,  and  'px^  -\-qx-{-r  be  the 
quotient,  and  let  Ax^  +  Bd^  -{-  Cx^  ■\-  l)x -{- E  be  the 
dividend. 

By  multiplying  the  divisor  and  quotient  together,  we 
obtain  as  coefficients  in  the  dividend, 

A        B            C  D          E 

ap       -hbp       +  cp  +  eg       -f  cr 

■i-aq       4-  &g  -\-br                >     ....     X 
-f  ar 

And  operating  upon  coefficients  only,  when  we  divide 
X  by   a  4-  6  +  c  we   should  get  p  -\-  q  -{-r]  or,  in  other 


DIVISION.  37 

words,  we  are  given  the  coefficients  in  X,  and  also  a,  6, 
and  Cj  and  we  are  to  obtain  p,  g,  and  r. 

Let  us  see  how  it  is  to  be  done : 

1.  Dividing  ap  or  ^  by  a  gives  p,  and  p  becomes  known. 

2.  Multiply  p  by  b,  and  subtract  the  product  from  B, 
leaving  aq.     Divide  aq  by  a,  and  we  have  q, 

3.  Multiply  g  by  6  and  p  by  c,  and  subtract  the  sum  of 
these  products  from  0,  leaving  ar.  Divide  ar  by  a,  and 
r  becomes  known. 

Thus  p,  q,  and  r  are  obtained. 

In  the  foregoing,  we  notice  —  (1)  that  the  only  quantity 
by  which  we  divide  is  a,  so  that  if  a  be  1  there  is  no 
real  division. 

(2)  If  we  change  the  signs  of  h  and  c,  the  partial 
product  hp,  bq,  br,  cp,  cq,  and  cr  all  become  additive,  so 
that  the  only  operations  involved  will  be  multiplication 
and  addition. 

(3)  That  the  partial  products,  which  form  any  co- 
efficient in  the  dividend,  as  (7,  are  made  up  by  a  cross- 
multiplication,  as  explained  in  Art.  34,  Ex.  1. 

This  process  is  known  as  synthetic  division,  because 
we  build  up  the  terms  of  the  dividend  by  getting  the 
partial  products  which  enter  into  their  composition, 
and  through  this  synthesis  we  obtain  the  terms  of  the 
quotient. 

The  following  examples  will  illustrate : 

Ex.  1.  To  divide    2a:*  +  x^  -  8x2  +  17x  -  12 
by  2x2-3x4-4. 


88 


THE  FOUR  ELEMENTARY  OPERATIONS. 


Here   a  =  2,   b  ■■ 
such  that — 


3,    c  =  4 ;    and  we  are  to  find  p,  g,  and  r 


■3, 


2p  =  2,     -3p  +  2g  =  l,     4j9-3g  +  2r: 
whence,     i?  =  1,  Q'  =  2,  r  = 

and  the  quotient  is  x^  +  2  cc  —  3. 

This  operation  is  systematically  carried  out  as  follows  — 


2  +  1-8 

3      6 

-4 


1  +  2- 


+    3-4 


Divisor,  signs  of  b  and  c  changed. 

+  17-12     .    .    Dividend. 

-  9  +  12^ 

-  8 


I .    .    Partial  products. 
0        0     .    .    Quotient. 


Here  we  change  the  sign  of  the  3  and  4  of  the  divisor, 
and  thus  have  only  additions.  We  then  divide  each  sum 
by  2  as  we  proceed.  Thus  the  multipliers  in  forming  the 
partial  products  are  3  and  —  4,  and  the  divisor  is  2. 


Ex.  2.   To  divide  x^  -  3  ic^  +  6  x  -  4  by  ^2  -  2  ic  +  1. 

The  coefficient  of  the  first  term  of  the  divisor  being  1  need  not 
be  written.     Making  the  functions  complete  in  form,  we  have 


1+0+0+0-3+0+0+0 

+2+4+6+8+4+0-4 

_1 -2-3-4-2+0 


1+2+3+4+2+0-2-4 


+  2- 


+  6-4 
-8  +  4 
+  2 


0      0 


and  the  quotient  is  x^  +  2  x^  +  3  x^  +  4  x*  +  2  x^  —  2  x  —  4. 

If  desired,  the  functions  may  equally  well  be  arranged 
in  ascending  order  of  the  variable,  as 


DIVISION. 


39 


-4+6+0+0+0-3+0+0 

^8-4+0+4+8+6+4 
+  4 -{_  2  +  0  -  2  -  4  -  3 


2-1 


+  0  +  1 
+  2  +  0. 
-2-1 


.4-2+0+2+4+3+2+1         0      0 
■  4  -  2  a;  +  2  x^  +  4  X*  +  3  x^  +  2  x"'  +  x^. 


40.  The  preceding  are  examples  of  exact  division.  In 
arithmetic,  when  the  dividend  is  greater  than  the  divi- 
sor, we  can  obtain  an  integral  quotient  and  a  remainder, 
where  there  is  one ;  or,  we  may  expand  the  remainder 
into  a  decimal  series  which,  in  general,  is  non-termina- 
ting. 

Now,  a  higher  degree  in  algebra  corresponds  to  a 
greater  quantity  in  arithmetic  ;  so  that,  when  the  divi- 
dend is  of  a  higher  degree  than  the  divisor,  and  the 
division  is  not  exact,  we  may  obtain  a  quotient  and  a 
remainder,  or  we  may  expand  the  remainder  into  an 
infinite  series. 

Ex.1.  To  divide  x^  -  x^  +  5x^  +  lOx^  -  5x  +  1  by  x*-2x8 
+  x^  —  2,  obtaining  the  quotient  and  the  remainder. 


+  2-    1+0  +  2    .    .    Divisor. 

1+0-1+0 

+2+4+4 

-1-2 

+  5  +  10-5  +  1     .    .    Dividend. 

+  4-    2  +  4  +  4^1 

-2+4                 ■ .    .    Partial  products. 

+  2 

1+2+2+2 
Quotient, 
x3  +  2x2  +  2a 

will  be  noticed  1 

+  9  +  12-1  +  5    .    .    Result. 

Remainder, 
,  +  2.               9x3+ 12x2 -x  + 5. 

bhat  a  vertical  line  is  drawn  to  the 

40     THE  FOUR  ELEMENTARY  OPERATIONS. 

left  of  that  part  of  the  divisor  which  is  used  in  forming 
the  partial  products. 

In  a  case  of  even  division,  all  the  terms  of  the  result 
to  the  right  of  this  line  are  zeros,  and  when  we  wish  to 
obtain  the  remainder  we  treat  these  terms  as  if  they 
were  zeros  in  forming  the  partial  products. 

If  we  employ  the  terms  to  the  right  of  the  vertical 
line  in  forming  partial  products,  the  quotient  will  extend 
into  a  series,  and  all  the  terms  to  the  right  of  the  line 
will  contain  negative  powers  of  a?,  and  the  series  will  thus 
be  arranged  in  descending  powers  of  x. 

If  we  wish  the  series  to  be  in  ascending  powers,  we 
must  arrange  our  functions  in  that  order  before  begin- 
ning the  division. 

Series  so  produced,  like  circulating  decimals  in  arith- 
metic, have  their  coefficients  connected  by  a  fixed  law 
of  formation.  Sometimes  this  law  is  obvious  from  sim- 
ple inspection,  and  at  all  times  it  can  be  exactly  deter- 
mined. 

This  law  is  of  great  importance  in  investigations  con- 
nected with  Recurring  Series. 

Ex.  2.  To  divide  1  +  a;  —  x^  by  1  —  2  x  +  x^  to  a  series  in 
ascending  powers  of  x. 


1 

+  2-1 

+  1-1 

+  2  +  6  +  8  +  10  +  12  H 

_1_3_    4_    5 

1 

+  3  +  4  +  5+    6+    7  +  ... 

Here  the  law  of  the  coefficients  is  obvious,  and  the  series  is 

1  +  3x  +  4x2  +  5x'^  +  6x*  +  7x5  +  ... 


DIVISION.  41 

If  we  arrange  the  dividend  and  the  divisor  in  descending  powers 
of  X,  the  quotient  coefficients  are  — 

-1-1  +  0+1  +  2  +  3  +  4  +  ... 
and  the  series  is  — 

/v  /y»0  /y^  rv*0  /y»0 

%Lf  JU  *K/  %K/  %Kf 

or  -  1  -  x-i  +  x-3  +  2  a;-*  +  3  x-5  +  4  x-^  +  ... 

Ex.  3.   To  divide  1  +  x  by  1  —  x  +  x^  to  a  series  in  ascending 
powers. 

The  series  is  1  +  2  x  +  x'^  -  x^  -  2  x*  -  x^  +  x^  +  ... 
Here  also  the  law  of  the  coefficients  is  readily  brought  out,  foi 
the  series  may  be  written 

(1  +2X  +  X2)(1  _X3  +  X6-X9+ ). 

Ex.  4.   Divide  1  by  1  -  2x  +  3x2. 


2-3 

1 

2  +  4  +  2-    8-22-20... 
_3-6-    3  +  12  +  33... 

1 

+  2  +  1-4-11-10  +  13... 

The  series  is  1  +  2x  +  x2  -  4x3  -  llx*  -  lOx^  +  13x6. ..,  and 
the  law  is  not  apparent  from  inspection, 

41.    The  following  results  are  frequently  required,  and 
should  be  committed  to  memory  : 

i.        -^  =  1  +  a:  -f  a;2  4-  a^  4-  a;4  _j 

ii.        — =— =  1  — a;4-ic2  — ic^-f-ic* 

l-{-x 

iii.    ^^        ^   =l4-2a;  +  3a^  +  4a^  +  5a;^+'" 


iv.   -— ± — ^  =  l_2a;4-3ic2-4ar»  +  5a;*- 


42     THE  FOUR  ELEMENTARY  OPEKATiOKS. 

14-2; 
Ex.  1.  To  expand    — — —  to  a  series. 

By  iii.  this  is  (1  ■}-  z)(_l  +  2 z  +  S z^  -\-  4 z^  +  ...),  which  by  dis- 
tribution becomes 

1  +  30  +  52:2  +  7^3+... 

Ex.  2.  To  find  from  what  division  the  series 

1  +  X  -  2x^  -  x^ -  X*  -h  2x^  +  x^  +  x^  -  2x^ "* 
has  been  derived. 

The  series  may  be  vsrritten 

(1  +  x-2x2)(l  -x^  +  x^-x^'"); 

and,  by  ii.,     1  —  x^  -\-  x^  —  x^  •••  = 

1  +  x3 

/.  1  +  a;  -  2  a;2  .^  ^^^  division. 
l  +  x3 

EXERCISE  II.  f. 

1.  Divide  x^  -\- x^ +  2x^ -2x  +  S  by  ic2  _  2 x  +  2. 

2.  Divide  6  a^^2  a^-\-9  a*-2  a^-a^-S  a-\-l  by  2a^+Sa-l, 

8.  Divide  1  +  x- +  2x^ -2x^  +  Sx^  by  1  +  2^  +  3x2  +  4x3. 

4.   Divide  a  —  lbya  +  lto5  terms  in  descending  powers  of  a. 

6.   Divide  x^  —  x^  -\-  x^  —  2x  —  1  by  x^  +  2 x2  —  3 x  +  1,  giving 
quotient  and  remainder. 

6.  Divide  a -\- 2a^  —  Sa^  -{-  4:a^'bj  a  —  a^  +  a^  —  a^  to  a  series, 
and  obtain  the  law  of  the  coefiicients. 

7.  If  ?/  =  1  -  0  +  (1  -  s)2  +  (1  -zy+  ...,  and  0  =  1  +  a:  +  a;2 
+  ic3  +  ...,  show  that  y  =—  x. 

8.  Divide  l+2x+3x2+4x3+...  by  l  +  3x+5ic2+7x3+...  to  6 
terms  of  a  series. 


DIVISION.  43 

9.  Expand  x  -i-  (x^  —  2  x -}-  1)  into  a  series,  first  in  ascending, 
and  second  in  descending  powers  of  x. 

10.  What  must  be  added  to  x'^  —  x^  +  x^  —  x  +  1  to  make  it  an 
even  multiple  of  x^  —  x  +  1  ? 

11.  Divide      1  +  fee  +  -\^-x'^  +  H^'  +  l«*  +  fl^^  +  ^ 
by  1 +  2x  +  3x2  +  4ic3 

12.  Divide  x^  +  j/^  _j_  3  ^.^^  _  1  l)y  a;  4. 2/  —  1. 
Take  x  as  variable  and  the  functions  are 

flj3  4.  0x2  +  32/x  +  (2/3  _  1)  and  x  +(y  -  1). 


1  +  0  +3?/ 


+  (y'  -  1) 
-  (y'  -  1) 


l-(2/-l)  +  (2/2  4-?/+l)  0 

Quotient  is  x^  -  x(y  -  1)  +  (y^  -\- y  +  1). 

13.  Divide   a^  (b  -  c)  +  b^  (c  -  a)  +  c^  (a  -  b)  hy  a  +  b  +  c. 
Take  a  as  variable  and  proceed  as  in  Ex.  12. 

14.  Divide   x^  +  2  x  ^^  -  x'^y^  +  a^?/*  -  2/^  by  x^  +  xi/^  _  y^. 

The  functions  being  each  homogeneous,  put  y  =  zx;  this  reduces 
the  division  to  x^  (1  +  2  z^  -  z^  +  z^  -  z^)  -4-  (1  -f  ^2  _  ^s), 

15.  Find  the  simplest  division  that  will  give  the  series 

X  +  3x2  +  2x3  -  x*  -  3x5  -  2x6  +  x7  +  ... 

16.  What  expression  added  to   (a  +  6  +  c)  (ab  +  5c  +  ca)  will 
make  it  exactly  divisible  by  a  +  5  ? 

17.  Multiply  a  +  3  +  3  a-i  +  a-2  by  1  -  a-i. 

18.  Multiply  (x  +  x-i)3  +  (x  +  x-i)2  +  (x  +  x-i)  +  1  by  x-x-i. 

19.  Divide  4  x2  -  7  x  +  3  -  x-2  +  x-3  by  x-i  -  2  x-2  +  x-^. 

20.  If  X  -  a  be  a  factor  of  x2  +  2  ax  —  3  b'^^  then  a  =  ±b. 


44     THE  FOUR  ELEMENTAKY  OPERATIONS. 

21.  If  1  -^  (a2  -  ax-\-  x^)  be  expressed  as  A  +  Bx+  Cx^  +  Do(? 
+  /(x),  find  A,  B,  O,  D  and  the  form  of  /. 

22.  If  x^  —  ax^  +  &X  +  c  be  divided  by  x  —  z^  the  remainder  is 
z^  —  az^  +  bz  +  c. 

23.  If  the  terms  of  the  divisor  are  written  in  an  inverted  order, 
by  what  arrangement  of  multiplication  are  the  partial  products 
formed  ? 


CHAPTER  III. 
Factors  and  Factorization. 

42.  In  a  case  of  even  division,  we  separate  the  dividend 
into  two  factors  (Art.  37).  One  or  both  of  these  might 
be  again  separated  into  two  factors,  and  so  on,  until  the 
whole  expression  was  separated  into  factors  which  should 
be  linear  in  the  variable  or  else  numerical. 

Thus      a^  ^bx^  —  a^x  -f-  a^b  =  (x  —  a){x-\-  a)  {x  —  b). 

We  thus  see  that  Factorization,  as  an  operation,  is  the 
inverse  of  Distribution. 

In  the  factored  expression  written  above  the  factors 
are  each  linear  and  binomial. 

In  6  a  (a  +  6  4-  c)  {ab  +  6c)  (oj^  +  £c  +  1),  6  is  a  numerical 
factor,  a  is  linear  and  monomial,  a-{-b  -{-c  is  linear  and 
trinomial,  and  ab  +  be  and  oc^  -^x  +  1  are  both  quadratic 
factors. 

Theoretically,  any  integral  function  of  a  variable  can 
be  separated  into  factors  linear  in  that  variable  ;  but  the 
cases  in  which  we  can  make  the  separation  practically 
are  limited  to  but  a  few  classes,  out  of  all  the  possible 
integral  functions.  Frequently,  however,  these  cases  are 
of  most  importance. 

43.  Factorization  may  sometimes  be  effected  by  mak- 
ing use  of  the  standard  forms  of  Art.  29. 

Thus,  because  a^  —  6^  =  (a  +  b)  (a  —  b),  we  can  always 

45 


46  FACTOKS   AND   FACTORIZATION. 

put  into  factors  that  which  can  be  expressed  as  the  dif- 
ference between  two  squares. 

Ex.1.     (a2  +  62)2_(a2_52)2 

=  (a^  +  62  +  «2  _  52)  (^2  ^_  52  _  ^2  4.  52)  ^  4  ^252. 
Ex.  2.  x'^-^2a-a^-l  =  x^-(a-  1)2=  (a;  -  a  +  1)  (x  +  a  -  1). 

44.  An  expression  of  the  form  x^  +  ax  +  b  can  be 
factored  at  sight  if  we  can  discover  two  quantities  such 
that  their  sum  is  a  and  their  product  is  b  whatever  a 
and  b  may  stand  for.  For  if  p  and  q  be  such  quantities, 
the  factors  are  {x  -\-p)(x-^q). 

Ex.1.   x'^-2x-S  =  (x-{-l)(x-S), 

Ex.  2.  a'^  +  2  ah  +  h'^  -  a  -  b  -  6  =  (a  +  5)2  -  (a  +  6)  _6 

=  (a  +  6-3)(a  +  6  +  2). 
Ex.3.   8ic2  +  8cc-6  =  2(2x*^  +  2.2x-3) 
=  2(2x  +  3)(2x-l). 

No  particular  rules  can  be  laid  down  for  this  kind  of 
factoring.  Success  is  to  be  attained  only  by  observation 
and  practice. 

EXERCISE  III.  a. 

1.  Put  (x'^  +  Sx  +  2') (x2  _  3 X  +  2)  into  four  linear  factors. 

2.  Put  ab  +  2a^  —  Sb^  —  ibc  —  ac  —  c^  into  linear  factors. 

3.  Express  x"^  —  (p -]- q)x^  +  pqx^  +  (p  -  q)x  —  1  as  the  pro- 
duct of  two  quadratics  in  x. 

4.  Factor  (x  +  yy  +  (x  -^  y)  (a  -]-  b) -^  ab. 

5.  Factor  (a  +  x)2  -  3(a  +  x)  +  2. 

6.  Factor  6(2 x  +  3 ?/)2  +  5(6x'^  +  bxy  -  ey^)- 6(Sx  -  2 yy. 

7.  Express  4a262 -(^2  +  52  _  ^2)2  in  the  form  of  four  linear 
factors. 


FACTORS   AND   FACTORIZATION.  47 

8.  Factor  12  ^2  -  7  x  +  1,  and  6  x^  -  21  x  +  18. 

9.  Express  x^  —  px^  +  (^  —  1)^^  -\-  px  —  q  as  two  linears  and  a 
quadratic. 

45.  Let  f{x)  be  any  integral  function  of  x,  and  let 
p,  q,  ?',  s,  etc.,  denote  its  factors,  so  that 

f{x)=p'q'r'S'" 

Now,  assuming  that  all  the  factors  are  finite,  if  any 
factor  becomes  zero,  the  whole  expression, /(a?),  becomes 
zero.  And,  conversely,  the  expression  cannot  become 
zero  unless  one  of  its  factors  becomes  zero. 

If,  then,  we  suspect  that  a  certain  expression  is  a 
factor  of /(cc),  we  put  that  expression  equal  to  0;  from 
this  we  find  the  value  of  x  in  terms  of  the  other  quanti- 
ties concerned,  and  we  substitute  this  value  for  x  in  the 
given  function. 

If  the  function  becomes  zero,  or  vanishes,  the  sus- 
pected expression  is  a  factor ;  and  if  the  function  does 
not  vanish,  the  suspected  expression  is  not  a  factor. 

Ex.  1.  Is  a  +  1  a  factor  of  6  d'h  +  3  a^  +  12  aft'^  -f  12  a6  +  3  a 
+  12  6^  +  6  6  ? 

Put  a  +  1  =  0 ;  this  gives  a  =  —  1.  Substitute  —  1  for  a  in  the 
given  function,  and  it  vanishes. 

Hence  a  -\- 1  is  a  factor. 

Similarly,  we  find  a  +  2  6  to  be  a  factor ;  and  by  dividing  by 
these  we  get  6  &  +  3  as  the  third  factor ;  and  the  expression 
becomes  (a  +  !)(«  +  2  5)(6  6  +  3). 

Ex.  2.   To  factor  a{h  +  he  -  c)+h{c  +  ca  -  a)+  c(a  -\-ab-  6). 

As  the  expression  may  have  a  monomial  factor,  try  a  =  0,  i.e. 
put  0  for  a. 

The  expression  vanishes,  and  hence  a  is  a  factor. 

But  the  expression  is  symmetrical  in  the  three  letters,  and 
hence  b  and  c  must  also  be  factors. 


48  FACTORS   AND   FACTORIZATION. 

Therefore  ahc  is  a  factor. 

By  the  index  law  the  dimensions  of  the  expression  must  be  the 
sum  of  the  dimensions  of  its  factors. 

But  ahc  is  of  three  dimensions,  and  so  also  is  the  expression ; 
hence  there  are  no  other  literal  factors. 

There  may  be  a  numerical  factor,  since  such  a  factor  has  no 
dimensions.  The  coefficient  of  the  type  ahc  from  the  expression  is 
readily  seen  to  be  3 ;  therefore  the  whole  expression  factors  to  3  ahc. 

Ex.  3.  To  factor  ah(h'^  -  a^)  +  hc(c^  -  62)  +  ca(a'^  -  c2). 

We  readily  discover  that  there  are  no  monomial  factors. 

Since  h  —  a  is  a  factor  of  one  of  the  terms,  let  us  try  if  it  be  a 
factor  of  the  whole. 

Put  h  —  a  =  0,  or  write  h  for  a  in  the  expression.     It  vanishes. 

Therefore  6  —a,  and  from  symmetry,  a  —  c,  and  c  —  6,  are 
all  factors. 

Therefore  (h  —  a)(a  —  c)(c  —  h)  is  a  factor  of  3  dimensions. 
But  the  expression  is  of  4  dimensions.  Hence  there  is  a  fourth 
factor,  symmetrical  in  a,  6,  and  c,  and  linear,  a  -{-  h  -\-  c  is  the 
only  such  factor  that  can  occur  here,  and 

(6  -  a)(a  -  c)(c  -  h)(a  +  6  +  c) 

is  a  factor,  and  comprises  all  the  linear  factors. 

For  the  numerical  factor  take  any  type  term  that  occurs  in  both 
the  expression  and  the  factored  result,  as  a'^h.  Its  coefficient 
from  the  expression  is  — ,  and  from  the  factored  result  it  is  +. 
Therefore  —  1  is  the  numerical  factor,  and  the  expression  becomes 

-(h-  a)(a  -  c^iic  -  h)(a  +  5  +  c), 
or  (a  -  6)  (5  -  c)  (c  -  a)  (a  +  6  +  c). 

Ex.  4.  To  factor  ah(c  -d)+  hc(d  -a)-\-  cd(a  -b)  +  da(h-  c). 

We  find  cf  to  be  a  monomial  factor,  and  by  symmetry  h,  c,  and 
d  are  factors. 

Therefore  abed  is  a  factor  of  4  dimensions. 

But  the  expression  is  of  only  3  dimensions,  and  should  have 
only  3  literal  factors,  unless  it  can  have  any  number  of  factors. 
The  only  expression  that  admits  anything  as  a  factor  is  0.  Hence 
the  expression  =  0,  identically. 


FACTORS   AND   FACTORIZATION.  49 

46.  If  we  have  an  integral  function  of  x,  and  if  its 
factored  form  be 

(a;  —  a){x  —  b)(x  —  c)  •  •  •, 

the  independent  term  of  the  function,  T  say,  is  equal  to 
abc'"  (Art.  31). 

Hence,  in  trying  to  factorize  the  function,  since  a,  b,  c, 
etc.,  are  all  factors  of  T,  the  factors  of  T  are  the  only 
quantities  to  be  substituted  for  x  in  our  trial. 

By  substituting  the  rational  factors  of  T,  any  rational 
linear  factors  of  the  function  will  be  discovered. 

Of  course  it  must  be  understood  that  factors  are  not 
necessarily  discoverable  in  this  way,  since  it  is  only  in 
special  cases  that  such  functions  have  all  or  any  of  their 
linear  factors  rational. 

Ex.  1.  To  factorize  x:^  -  Sx^  -  Sx^  +  7 x  +  6. 

Here  T  =  Q,  and  its  factors  are  ±1?  ±2,  ±3,  and  ±  6. 

Put  1  for  x;  the  function  does  not  vanish,  and  x  —  1  is  not  a 
factor. 

Put  —  1  for  X  ;  the  function  vanishes  and  x  +  1  is  a  factor. 

Similarly,  try  2,  —  2,  3,  —  3,  etc.,  successively  until  all  the  fac- 
tors discoverable  by  this  means  are  found. 

Otherwise,  having  found  x  +  1  to  be  a  factor,  divide  the  func- 
tion by  X  -\-  1.     The  quotient  is 

x^-ix^  +  x  +  6. 

Of  this  new  function  x  +  1  is  again  a  factor.  Divide  by  x  4-  1, 
and  the  quotient  is  2      ^      i  a 

which  factors  into  (x  —  3)  (x  —  2) . 

.-.  X*  -  3x3  -  3x2  +  7x  +  6  =  (x  +  l)2(x  -  3)(x  -  2). 

In  employing  this  latter  method,  it  will  be  more 
expeditious  to  try  the  higher  factors  of  T  first. 


50  FACTOKS   AND   FACTORIZATION. 

EXERCISE  III.  b. 

1.  Factorize  the  following  — 

i.  2a25  +  2a5c.  Iv.   2{a(65-c3)}. 

ii.  ^{ah'^-aT-h).  v.    ^{h{c  -  d){ah  -  cd)}, 

iii.  S(a.&^-c-^).  vi.    S{a(6't-cO}. 

vii.  2  (a62  _  a'^j^)  +  Sa&  -  2^2  +  i. 

viii.  (a+ 6 -c)(?)  + c- a)(c  + a- &)- 2a6(a  + 6)  +  2a^     ° 

ix.  2  {ah  '  c  —  d)  four  letters. 

2.  Put  2a*6c  —  2a^63  j^to  quadratic  factors. 

3.  If  the  6th  power  of  a  number  be  diminished  by  1  and  the  5th 
power  of  the  same  number  be  increased  by  1 ,  the  difference  of  the 
results  is  divisible  by  the  next  greater  number. 

4.  If  any  even  power  of  a  number  be  diminished  by  1,  and  any 
odd  power  of  the  same  number  be  increased  by  1,  the  results  have 
a  common  factor. 

5.  Factorize  x*  +  8^3  -  10x2  -  104  x  +  105. 

6.  Express  (x  +  l)(x  +  3)(x  +  5)(x  +  7)  +  15  as  the  product 
of  two  linears  and  a  quadratic. 

7.  Factorize  x*  —  5  x2  +  6. 

8.  Factorize  (a+6  +  c)3-(5  +  c-a)3-(c+a-5)3-(a+6-c)3 

9.  Factorize  ^a{bc  +  ac  +  d^  —  a^}  —  5  ahc. 

10.  Factorize 

{ac  +  hdy^—  abc{a  —  5  4-  c)  —  hcdQb  —  c  -{■  d)—  cda(c  —  d  +  a) 
—  dab{d  —  a  +  6). 

11.  Factorize 

(a  +  &  -  x)  (5  +  c  +  x)  +  (6  +  c  -  x)  (c  +  a  +  a;) 
+  (c  +  a  —  x)  (a  +  ^  +  ic)  —  3(a5  4-  5c  -f  ca) 
4-  ahc  -  a2  _  52  _  c2  _^  3  x'^. 


FACTORS   AND   FACTOR IZATION.  5l 

12.  Factorize 

{a  ~  h){h  +  c)+{h  -  c)(c  ^  a)  +  {c  -  a) {a  +  &)  +  («-  c)2. 

13.  Factorize  x^  +  x{^ah  -  Sa^) -(^a-h){h-  c) (c  -  «). 

14.  If  the  4th  power  of  a  number  be  increased  by  4  and  dimin- 
ished by  5  times  the  square  of  the  number,  the  result  multiplied  by 
the  number  itself  is  the  product  of  5  consecutive  numbers. 

47.  The  symbol  ^  is  defined  by  the  relation  -yja  x  V^ 
=  a,  where  a  denotes  any  numerical  quantity  or  algebraic 
expression. 

Thus     V(^^  +  &'  +  2a6)  =  a  +  &,  and  V^^  ^  4. 

The  expression  ^a  is  read  the  square  root  of  a,  or,  more 
concisely,  the  root  of  a,  and  it  denotes  that  a  is  to  be 
separated  into  two  identically  equal  factors,  and  that  one 
of  these  is  to  be  taken. 

Hence  when  it  is  possible  to  separate  a  quantity  or  an 
expression  into  two  such  factors,  it  is  possible  to  express 
exactly  the  square  root  of  the  quantity  or  expression. 

Thus  Sa^  +  2  ^ah  can  be  separated  into  the  identically 
equal  factors  la  x  2a;  and  hence  %a -^ -yj {%a^  -\- 2^ab) , 
for  any  number  of  letters. 

48.  The  expression  y'a  requires  careful  consideration. 

(1)  If  a  is  a  positive  square  number  it  is  the  product 
of  two  identically  equal  factors,  and  ^a  denotes  one  of 
these  factors. 

The  factors  may  be  both  +  or  both  — ,  since  in  either 
case  the  product  is  -f . 

Therefore  ^a  has  two  signs  and  is  often  written  ±  -y/a. 
Thus  V^  is  either  +  x,  or  —  a? ;  and  ->/16  is  +  4,  or  —  4. 

The  double  sign,  whether  written  or  not,  must  always 


52  FACTORS   AND  FACTORIZATION. 

be  mentally  attached  to  a  square  root,  being  frequently 
of  very  great  importance. 

(2)  If  a  is  a  positive  non-square  number,  no  two  iden- 
tically equal  factors  can  be  found  for  it. 

The  -y/a  then  symbolizes  one  of  that  class  of  numerical 
quantities  called  incommensurables,  or  irrational  quanti- 
ties (Art.  1). 

In  this  case  ^a  cannot  have  its  value  exactly  expressed ; 
but  it  may,  under  the  form  of  a  non-terminating  decimal, 
be  expressed  to  any  degree  of  approximation  we  please, 
by  the  arithmetical  process  of  ^extracting  the  square 
root.' 

Thus  (1.41)2  differs  from  2  by  0.0118 


(1.414)2 

n 

"     2  "  0.000604 

(1.4142)2 

i( 

«  2  "  0.000039 

(1.41421)2 

(( 

"  2  "  0.0000002 

etc. 

etc. 

And  the  successive  squares  become  closer  and  closer- 
approximations  to  2,  the  degree  of  approximation  de- 
pending upon  the  extent  of  the  decimal  series. 

This  series,  unlike  circulants  produced  by  division, 
has  no  arrangement  of  its  digits  which  would  indicate 
any  law  governing  the  order  of  their  succession. 

(3)  Let  a  denote  a  negative  number. 

As  like  signs  produce  only  -f  in  multiplication,  it  is 
not  possible  to  find,  or  to  approximate  to,  or  to  conceive 
of  a  quantity,  which  multiplied  once  by  itself  will  give 
the  sign  — . 

The  symbol  ^a  is  then  called  an  imaginary  in  contra- 
distinction to  the  quantities  of  (1)  and  (2),  which  are  real. 

Thus  "y/—  3  is  imaginary,  while  -y^3  is  real. 


FACTORS   AND   FACTORIZATION.  53 

49.  As  in  arithmetic,  so  in  algebra,  an  expression  may 
be  a  complete  square  and  be  capable  of  having  its  square 
root  exactly  expressed,  or  it  may  be  a  non-square  and 
admit  only  of  having  its  root  approximated  to  by  an 
infinite  series. 

Thus 


1      x"  ,  I'S    a^      1.3.5    a?^ 


^  ^2     1.2    22^1.2    2^     1.2.3   2^^ 

General  methods  for  this  approximation  will  be  con- 
sidered hereafter. 

It  may  be  remarked  that  an  algebraic  expression  can- 
not in  itself  be  imaginary,  as  the  character  of  real  or 
imaginary  is  wholly  due  to  the  interpretation  of  the 
quantitative  symbols. 

Thus  Va  —  6  is  real,  if  a  and  h  are  both  positive 
numbers,  and  b  is  less  than  a,  but  imaginary  if  5  is  greater 
than  a.  If  a  is  positive  and  b  negative,  the  expression 
is  always  real ;  and  if  a  is  negative  and  b  positive  it  is 
always  imaginary. 

50.  If  a  denotes  a  positive  quantity,  Va  can  be 
expressed  to  any  degree  of  approximation  that  we  please, 
and  hence  ^a  must,  like  other  quantitative  symbols,  be 
subject  to  the  commutative  and  distributive  laws. 

Hence 

(1)  V^+  ^b=^b-\--y/a,  and  ya  •  V^=  V^ '  V^- 

(2)  ■y/a(b-{-  V^)  =b^a  +  ^a-  ^c,  etc. 

51.  The  symbol  y'  is  introduced  here  as  a  special 
operative  symbol,  having  a  relationship  to  the  exponent, 
as  will  appear  hereafter,  and  it  is  necessary  that  we 
should  investigate  the  limits  of  its  operation,  and  dis- 


54  FACTOKS   AND   FACTOHIZATION. 

cover   in  how  far  it  obeys  the   great   formal   laws   of 
algebra. 

(1)  As  (ya- V^)(V^*  V^)  =  V<^*  V^*  V^' V^  =  ^^ 
by  definition  ;  and  as  Va6  •  -\/ah  =  ah  by  definition, 
therefore  ^a  •  ^h  =  y/ah ;  and  the  operative  symbol  ^ 
is  distributive  over  the  factors  of  a  prodxict. 

Thus     V^  •  V^  •  "^^  =  V(4 ^^^)  ^  V^  •  Va6  =  2  Va6. 

V (45 a^ &c2)  =  V (^ ^'c' .  5 a&)  =  3 ac  V5a6. 
etc.  etc.  etc. 

(2)  Since  ^{a-\-b)^(a-{-b)  =a -\-b  by  definition, 
and  (-^a  +  V^)  (ya  +  -^b)  =a-\-b-{-2  -y/ab  by  distribu- 
tion, therefore  ->/(a  +  6)  is  not  the  same  as  ->/a  +  V^* 

Or,  the  symbol  ^  is  not  distributive  over  the  terms  of  a 
sum. 

The  statements  of  (1)  and  (2)  form  the  working  prin- 
ciples of  this  symbol,  and  should  be  carefully  remembered. 

52.  Since  ^— a-^— a  =  —  a  by  definition,  whatever 
a  may  be,  we  assume  that  ^a  is  subject  to  the  same 
general  laws  of  operation  whether  a  be  positive  or  nega- 
tive, i,e,  whether  the  expression  be  real  or  imaginary. 

■  a  .  -Vb  =  y'a  •  V—  bj 

where  ^a  is  real. 

Thus  every  imaginary  number  can  be  reduced  to  depend 
upon  the  symbol  V—  1,  which  is  called  the  imaginary 
unit,  and  is  usually  symbolized  by  i. 


FACTORS   AND   FACTORIZATION.  55 

If,  then,  X  denotes  any  real  number,  ix  denotes  the 
corresponding  imaginary ;  the  relation  between  these 
being  that  the  square  of  the  first  is  +  a^,  and  of  the 
other  it  is  —x\ 

This  generalization  introduces  us  to  a  new  set  of  num- 
bers, the  symbolic  numbers  or  imaginaries. 

All  whole  numbers,  positive,  negative,  and  imaginary, 
may  be  represented  in  the  general  scheme, 

..._5,    -4,    -3,    -2,  -1,0,1,2,   3,   4,  5,   ... 
...  —  ^i^  —  4 1,  —  3 i,  —  2i,  —  ij  0,  i,  2i,  3i,  4i,  5 i,  ••• 

53.  The  powers  of  the  symbol  i,  which  occur  very 
frequently,  are  given  in  the  scheme, 

i^  =  —  1,  i^  =  —  i,  i*  =  1,  f  =  i,  i^  =  —  1,  etc., 

the  powers  repeating  their  values  in  cyclic  order. 

A  quantity  which  is  the  algebraic  sum  of  a  real  and 
an  imaginary  is  called  a  complex  quantity;  but  being 
arithmetically  inexpressible  on  account  of  its  imaginary 
part,  it  ranks  with  imaginaries. 

Thus,  2  +  31  is  a  complex,  and  so  also  is  2  —  3  i.  The 
square  of  2  +  3i  is  2^+  (3i)^4-12i,  or  12 z  — 5,  another 
complex ;  but  the  product  (2  +  3  i)  (2-3  i)  is  2^  -  (3  i)  \ 
or  13,  a  real. 

a  +  hi  represents  any  number  whatever;  for  if  6=0,  it 
is  real ;  if  a  =  0,  it  is  imaginary ;  and  if  neither  a  nor  h 
be  zero,  it  is  a  complex. 

64.  The  expression  x^  -\-px-\-q  can  always  be  sepa- 
rated into  two  factors  linear  in  x. 


56  FACTORS   AND   FACTORIZATION. 


=^(x  i  P  +  ^P'-'^^Vx  I  p-Vp'-4g 

As  these  factors  both  contain  the  same  square  root 
part,  they  will  be  both  real  or  both  complex,  i.e.  imaginary. 
But  in  any  case  they  will,  upon  distribution,  reproduce 
the  original  expression. 

Thus  the  factors  of  a;^  +  2  a;  +  3  are 

both  being  complex  quantities. 

55.  The  expression  aa^  -{-bx  +  c  can  always  be  sepa- 
rated into  a  monomial  factor,  a,  and  two  factors  linear  in  x, 

aoc^  +  bx  +  c^a{x^-{--x-\-- 
\        a        a 

The  part  within  brackets  is  the  same  expression  as 

ar'  -f-  »aj  +  g,  if  we  write  -  for  p  and  -  for  q. 
a  a 

Making  this  substitution  in  the  result  of  Art.  54,  gives, 
after  reducing, 

x-\ — 


«{-^±tf^}{' 


2a 


This  factorization  may  also  be  done  independently  as 
follows : 

aa^  +  2>a;  +  c  =  —  { 4  a  V  +  4  abx  +  b^ -(b^ -4.ac)\ 

=  — K2aa;  +  &)2-(V62-4ac)2| 
4a  ^  ^ 


FACTORS   AND  FACTORIZATION.  57 

==^l2ax-^b-\-Vb''-4.ac\l2ax-\-b-^/b^-4.ac\ 

Ex.  1.   The  factors  of  3ic2  +  5  a;  -  1  are 


Ex.  2.    The  factors  of  2  ^2  -  3  x  +  2  are 

The  two  factorizations,  of  Art.  54  and  the  present 
Article,  are  very  important,  and  the  forms  of  the  factors 
should  be  carefully  mastered. 

Taking  ax^  -{-bx  +  c  as  being  the  most  general  in 
form,  the  square  root  part  of  each  factor  is  -Vb^  —  4  ac. 
The  character  of  the  factors  will  depend  upon  that  of 
this  part  of  them. 

If  a  and  c  are  +,  and  4ac  is  greater  than  6^,  the  fac- 
tors will  be  complex  quantities. 

If  4  ac  be  less  than  6^,  the  factors  will  be  real  and 
unequal ;  and  if  4  ac  =  b^,  the  expression  V6^  — 4ac 
becomes  zero,  and  the  factors  are  real  and  equal. 

Cor.  The  expression  aa^  -{-bx-\-c  is  a  complete  square 
when  6^  =  4  ac. 

The  finding  of  the  square  root  of  an  algebraic  expres- 
sion is  equivalent  to  the  separation  of  the  expression 
into  two  identical  factors,  and  hence  requires  no  special 
process. 

If  the  expression  is  a  complete  square,  the  factorizar 


58  FACTORS   AND   FACTORIZATION. 

tion  ranks  with  the  simpler  cases.  But  if  it  is  not  a 
complete  square,  the  only  practical  method  is  by  means 
of  the  binomial  theorem  or  undetermined  coefficients,  to 
be  given  hereafter,  and  the  root  is  expressed  as  an 
infinite  series. 

Thus,  in  a^  —  2  a&  +  a  +  4  5^  —  4  &  +  1,  we  readily  see 
that  ±  a,  ±2b,  and  ±  1  must  be  terms  of  the  root,  and 
the  least  observation  shows  that  the  root  is  a  —  2  &  -|- 1. 

Ex.  3.   The  factors  of  2  x^  —  3  x  —  4  are 

2{x  +  }(-3+ V41)}{x  +  K-3-V41)}. 

Ex.  4.    The  factors  ot  x^  +  2x+10  are 
(X  +  1  +  Si)(x  +  1-Si). 

Ex.  5.  Determine  the  value  of  m  so  that  3  x^  +  4  mx  +  12  may 
be  a  complete  square.     We  must  have 

62  =  4  ac,  i.e.  (4  m)2  =  4  •  3  •  12. 

Whence  m  =  4-  3  or  —  3. 

56.  The  expression  x"^  -\-  bx^  -\-c  can  be  separated  into 
four  linear  factors.  Let  a,  /?,  y,  S  denote  these  factors. 
Any  two  of  these  multiplied  together  gives  a  quadratic 
factor  of  the  expression. 

But  these  may  be  multiplied  two  together  in  three 
different  ways,  namely, 

afi,  yS ;  ay,  )8S  ;  and  a8,  /?y. 

Hence  x'^  -{-bx^  -\-  c  can  be  separated  into  a  pair  of 
quadratic  factors  in  three  different  ways. 

Ex.  x*  +  10x2  +  9=  (ic2  +  9)(x2  + 1) 

=  (x  +  Si)(x  —  S  i)  (x  +  i)  (x  —  i). 


FACTORS   AND   FACTORIZATION.  59 

Thence  the  pairs  of  quadratic  factors  are  — 

1.  (a;2  +  9)(a:2  +  l). 

2.  (x^  -\-iix-  3) (x2  -iix-S), 

3.  (^2  +  2ix-\-  3)  (ic2  -  2 IX  +  3). 

57.    The  factorization  of  x^  —  1  is  important, 
and  by  Art.  54, 

If  any  one  of  these  factors  becomes  zero,  the  expres- 
sion vanishes ;  i.e.  ar^  —  1  =  0,  and  qc^  =  1. 

Hence,  when  x  =  l,  or  -  ^^W^  or  -  hth/^  a^  =  1. 
'  '  2  2 

And  since  the  cube  of  each  of  these  three  values  of  x 
is  1,  these  are  the  three  cube  roots  of  unity j  one  of  which 
is  real,  and  the  other  two  complex. 

The  complex  roots  are  generally  denoted  by  <o  and  w^, 
because  the  square  of  either  of  them  is  equal  to  the 
other. 

Then  w^  =  1,  w*  =  w^ .  w  =  w,  w^  =  o)%  w^  =  1,  etc. 

Since  w^  — 1  =  0,  its  equivalent  (cd— l)((o^4-  a)-f-l)=0; 
and  as  0)  —  1  is  not  zero,  we  must  have 

0)2  +  a>  +  1  =  0. 

And  this  is  the   fundamental   relation   connecting   the 
cube  roots  of  1 ;  i.e.  the  sum  of  the  three  roots  is  zero. 


60  FACTORS   AND   FACTORIZATION. 

Ex.  Multiply  together  x  +  wy  +  w^^  and  x  +  w^y  +  wz. 
Distributing  (x  +  wy  +  ta^z)  {x-\-  (ohj  ■\-  wz)  gives 

But  w3  =  1,  a»2  +  <u  =  -  1,  and  w*  +  (o2  =  w  +  w^  =  -  1. 

/.  (x  +  wy  H-  w2;j) (x  +  w2?/  +  wz)  =:x'^  +  y'^  ■\-  z"^  —  xy  —  yz  —  zx. 

EXERCISE  III.  c. 

1.  Find  V{«^  +  62  +  2  a6  -  2  a  -  2  6  +  1}. 

2.  Separate  x2  +  4y2^4a.2^_4ic_8?/4-4  into  two  identically 
equal  factors. 

3.  Express  Vab^  +  Vac*  —  ^74  aft^^^  as  a  multiple  of  a  single 
irrational  factor. 

4.  Show  that  (a  +  hi)  (c  +  di)  has  the  form  A  +  J5i,  and  ex- 
press A  and  B  in  terms  of  the  small  letters.    • 

5.  Distribute  (x  —  a  +  6Q  (x  —  a  —  ?>i) . 

6.  Show  that  ix  -  I  (ix)2  +  J  {ixY  —  \  (zx)*  + 

=  i(x-Jx8+ix5-+...)+ix2(l-i^'+J^*-+-). 

7.  Factorize  the  following  — 

1.  x2-3x-3.  V.   ax2  +  x4-a. 

ii.   x2  -  2  X  +  5.  vi.  px'^  -  (p  +  1)  x  +  1. 

iii.  4  x2  -  4  X  -  2.  vii.    {a^  -  62)  x2  +  2  ax  +  1. 

iv.    5  x2  +  3  X  -  6.  viii.  2  +  ax  +  — • 

ax 

8.  Resolve  x*  —  11  x2  +  10  into  four  linear  factors. 

9.  Resolve  x*  —  3  x2  -f  1  into  linear  factors. 

10.  Resolve  x*  +  x2  —  2  into  its  three  pairs  of  quadratic  factors. 

11.  Resolve  4x2  —  4x  —  x*  +  l  into  linear  factors. 

12.  Resolve  x^  —  2  x2  —  x  +  2  into  linear  factors. 

13.  Distribute  (1  +  x)(l  +  t«^x)(l  +  w2x). 


FACTORS   AND   ROOTS.  61 

14.  Show  that  (a  +  w/3  +  ^27)  (a  +  (o^^  +  W7)  =  Sa^  -  Sa/3. 

15.  Show  that  {x-\-  tay  -\-  uiHf  +  (a;  +  ^^2/  +  w^)^ 

=  2  2x3 -3  2x2?/ +  12x2/0. 

16.  Distribute  (x  +  w?/  +  w2^)  (x  +  0^22/  4-  iaz)  (x  +  2/  +  «). 

17.  Show  that  (x  +  w?/  +  ^2^)3  -  (x  +  0*22^  +  ^^y 

=  -  3  V^=r3  (X  -  2/)  (2/ -  2;)  (;?  -  X). 

18.  Find  the  relation  between  a  and  6  when 

(a  +  &)x2  -  (a2  -  62)x  +  a26  +  aW- 
is  a  complete  square. 

58.    The  integral  function  of  a?, 

ic*  _  lOa^  +  35a;2  -  50a;  +  24, 

factors  into 

(a;  _  1)  (a;  _  2)  (a;  -  3)  (a;  -  4). 

The  numbers  1,  2,  3,  4  are  the  roots  of  the  function, 
because  from  these,  and  the  variable  x^  we  may  build  up 
the  function  by  multiplication,  or,  so  to  speak,  cause  it 
to  grow  up. 

If  any  of  these  roots  be  put  for  x,  the  substitution  will 
cause  the  function  to  vanish,  since  it  makes  one  of  the 
factors  zero.  And,  conversely,  the  only  single  substitu- 
tion that  will  make  the  function  vanish  must  make  one 
of  the  factors  vanish,  or  is  the  substitution  of  one  of  the 
roots  for  x. 

Hence  the  roots  of  an  integral  function  of  any  variable 
are  those  quantities  which,  when  put  for  the  variable  in 
the  function,  cause  it  to  vanish.  And  reciprocally  any 
quantity,  which  put  for  the  variable  will  cause  the  func- 
tion to  vanish,  is  a  root. 


62  FACTORS   AND   ROOTS. 

Thus  the  roots  of  a^  +  x^'^a  +  x^ab  +abc  are  —  a,  —  b, 
and  —  c  J  for  the  function  factors  into 

(x  +  a)  (ic  +  b)  (x  +  c). 

59.  The  expression  a;*  —  10  ic^  +  35  ic^  —  50  a?  +  24  =  0  is 
a  conditional  equation,  or  simply  an  equation  in  which  a; 
is  to  have  such  a  value  as  will  make  the  expression  an 
identity  (Art.  22). 

We  have  seen,  in  the  preceding  article,  that  this  will 
be  effected  by  making  x  equal  to  any  one  of  the  roots  of 
the  function,  namely,  1,  2,  3,  or  4. 

It  is  readily  seen  that  the  same  principle  applies  to 
fimctions  of  any  degree. 

Hence:  (1)  In  the  equation  formed  by  putting  an 
integral  function  of  a  variable  equal  to  zero,  we  obtain 
the  roots  of  the  equation  by  separating  the  function  into 
factors  linear  in  the  variable. 

The  determination  of  any  one  of  these  factors  is  a 
solution  of  the  equation,  and  the  determination  of  all 
these  factors  is  the  complete  solution. 

(2)  The  whole  number  of  solutions,  or  the  number  of 
roots  which  the  equation  has,  is  the  number  of  linear 
factors  into  which  the  function  is  theoretically  separable, 
and  this  is  the  same  as  the  degree  of  the  function  in  the 
variable. 

The  solution  of  an  equation  is  thus  equivalent  to  the 
factorization  of  the  function  into  factors  linear  in  the 
variable. 

60.  When  the  roots  of  an  integral  function  or  of  the 
corresponding  equation  are  all  real  and  all  rational,  they 
can  generally  be  found. 


FACTORS   AND   ROOTS.  63 

Also,  the  methods  of  factoring  now  at  our  disposal  are 
sufficient  for  the  linear  factorization  of  all  integral 
functions  of  a  single  variable  of  not  more  than  two 
dimensions ;  but  these  methods  are  not  sufficient  for 
the  general  factorization  of  functions  of  more  than  two 
dimensions.  They  suffice,  however,  for  many  special 
and  particular  cases. 

Ex.  1.    To  find  all  the  solutions  of  x*  -  x^  -  2  ic2  -  2  +  4  =  0. 
By  trial  we  readily  find  x  —  I  and  ic  —  2  to  be  factors. 
Dividing  by  these  gives  ic^  +  2  aj  +  2  =  0. 
The  factors  of  this  are  (x  +  1  +  i)  (x  +  1  —  i) . 
And  the  four  roots  are  1,2,  —  1  —  ^  and  —  1  +  i. 

Ex.  2.    To  solve  the  equation  x^  —  bx-\-2  =  0. 

Factorization  gives  (x  —  2)  (x^  +  2  a;  —  1). 

The  factors  of  ic2  +  2  x  -  1  are  {x  +  (1  +  ^^)}{x  +  (1  -  V^)}- 

And  the  roots  are  2,  -  (1  +  ^2),  -  (1  -  V^)- 

A  linear  equation  in  any  variable  is  simply  a  linear 
factor   of  unity.     By   proper  transformations   such   an 
equation  may  always  be  brought  to  the  form 
Ax-B  =  0, 

B 

or  ^=   .' 

A 

which  is  the  only  solution,  x  having  but  a  single  value. 

EXERCISE  III.  d. 

1.  Solve  the  following  equations  — 

1.   x^  +  4  x^  —  X  —  4  =  0. 

ii.  x5  +  x^  -  x-2  -  1  =  0. 
iii.   x3  +  3x2  +  4x  +  2  =:  0. 

2.  Find  values  of  x  that  will  make  (&2  _  a2)x  +  ^^^  -^  equal 
to  (x-l)(a-6).  ^ 


CHAPTER   IV. 

Highest  Common  Factor.  —  Least  Common 
Multiple. 

61.  The  expressions  2a^bc  and  6ab^  have  2,  a,  and  b  as 
factors  common  to  both,  and  the  product  of  these,  2aby 
is  the  highest  common  factor  of  the  expressions. 

The  name  Highest  Common  Factor  is  contracted  to 
H.  C.  F.,  and  sometimes  G.  C.  M.  (greatest  common 
measure)  is  used  in  its  stead. 

The  expressions  ar^  —  7 ic  +  6  and  a;*  +  2a^— 9a^  +  8 
factor  respectively  into  {x  —  l){x  —  2)  {x  +  3)  and 
{x  —  1)  {x  -\-l)  {x  —  2)  {x  -{-  4t).  They  accordingly  have 
the  binomial  factors  (cc  —  1)  (x  — 2)  or  ic^  — 3a;  +  2  as 
their  H,  G.  F, 

Common  monomial  factors,  where  they  exist,  are  readily 
detected  by  inspection.  To  detect  binomial  factors,  we 
may  factor  the  expressions  and  pick  out  the  common 
factors,  as  in  the  preceding  example,  or  we  may  proceed 
upon  the  principle  now  to  be  established. 

62.  Theorem.  If  two  expressions  have  a  common 
factor,  the  sum  and  the  difference  of  any  multiples  of 
the  expressions  have  the  same  common  factor. 

Let  A  and  B  denote  the  two  expressions,  and  let  / 
denote  their  common  factor,  so  that  A  =  Pf  and  B  =  Qf, 
where  P  and  Q  denote  all  the  factors  remaining  in  A 
and  B  respectively  after  the  removal  of/. 
64 


HIGHEST   COMMON  FACTOR.  65 

Let  a  and  h  be  any  numerical  multipliers. 

Then,     aA  ±bB  =  aPf±  bQf=  (aP  ±  bQ)fi 
and  as  this  last  expression  contains  the  common  factor 
/,  the  theorem  is  proved. 

63.  Now  let  A  and  B  be  two  integral  functions  of  x, 
and  let  them  have  the  common  factor  /,  which  we  will 
suppose  to  be  quadratic,  as  their  H.  C.  F. 

S J  taking  the  sums  or  differences  of  proper  mu.tipies 
of  A  and  B,  Ave  may  reduce  the  dimensions  of  each  by 
unity,  and  obtain  two  new  functions  A^  and  J5',  one 
dimension  lower  respectively  than  A  and  B,  and  of 
which  /  is  still  a  common  factor. 

By  operating  in  a  similar  manner  upon  A^  and  B\  we 
find  two  functions  A^^  and  .B",  two  dimensions  lower 
respectively  than  A  and  B,  and  containing/  as  a  common 
factor. 

,_  By  a  continuation  of  this  process  we  must  eventually 
reduce  A  and  B  to  depend  upon  functions  of  two  dimen- 
sions, and  having  /  as  a  common  factor. 

Hence  these,  upon  rejecting  all  monomial  factors,  must 
be  the  factor  /,  and  must  therefore  be  identical. 

And  thus  the  identity  of  the  two  results  at  any  stage 
of  the  operation  indicates  that  the  H.  C.  F.  is  obtained. 

Ex.1.    Let  ^  =  6x3-7x2-9^-2,  B  =  2x^+Zx'^-\lx-Q. 

Operation. 

u4  ....  6x3-7x2-    9x-    2       B    ...  .2x3+    3x2-  llx-6 

35   .  .  .  6x3  +  9x2-33x-  18       3^.  .  .  ISx^  -  21x2  -  27x  -  6 

ZB-A    .  .  .  16x2-24x-16       3^-5   16x3  -  24x2  -  16  x 
Reject  factor  8,      2 x2  —  3 x  —  2      Reject  factor  8x,   2x2  —  3x  —  2 
The  results  being  identical  shows  that  the  highest  common  fac- 
tor is  2x2 -3x- 2. 


66  HIGHEST  COMMON  FACTOR. 

64.   In  the  preceding  example  we  notice : 

(1)  That  the  presence  of  the  variable  is  unnecessary, 
and  the  operation  may  be  carried  out  upon  the  coefficients 
alone. 

(2)  That  as  we  reject  monomial  factors  wherever  they 
occur,  all  monomial  factors  should  be  removed  before 
beginning  the  operation,  and  that  any  of  these  that  are 
common  to  both  functions  should  be  set  aside  and  be 
multiplied  by  the    final   result   to   give    the    complete 

H.  a  F, 

Ex.  1.   Let  ^  =  3rK3-10ic2-|-9a;-2,  5  =  2a;8  -  7«2  +  2«  +  8. 

A 3-10+9-2      B 2-      7+      2  +  8 

2  A 6-20+18-4      4^ 12-40+36-8 


ZB  .  .  .  , 

.  .  .  6  -  21  +    6  +  24 

A'  .  .  ,  . 

19A'   .  .  , 
14^'   .  . 

1  +  12-28 

.  .  .  .   19  +  228  -  532 

.  ...  196  -  658  +  532 

B' 14  -    47  +    38 

14^'  ....  14  +  168-392 


A" 215-430 

-f-215 .1-2 


B" 215-430 

-^215 1-2 

.-.  x-2  is  the  H.  C.  F. 


Ex.2.    Let  J[  =  3x*-6x3-6x-3,   5  =  6x4  +  12x3  +  12-6. 

3  being  a  common  monomial  factor,  we  set  it  aside  and  write 

A 1-2  +  0-2-1       B 1+2  +  0  +  2-1 

B 1  +  2  +  0  +  2-1       iA' 1  +  0+1 

A' 4  +  0  +  4  B' 2-1  +  2-1 

iA 2  +  0  +  2 


-4 1  +  0  +  1 

.-.  3(x2  +  1)  is  the  H.  C  F. 


B" 1  +  0  +  1 


HIGHEST   COMMON  FACTOR.  67 

Ex.  3.   Let  ^  =  3x4-55c3  +  5c2  +  4x+l,  B  =  Sx^-nx-4. 

A 3-5+1+4  +  1      B 3-11-4 

B 3-11-    4  4  A" 108  +  48  +  4 

B" 111  +  37 

-f-  37,  B'" 3  +  1 

.-.  Sx  +  lis  the  H.C.F. 


A'   .  .  ,  . 
2B 

.  .  .  6+  5+  4  +  1 
...  6  -  22  -    8 

Af 

9B'"  .  .  . 

27  +  12  +  1 

27  +    9 

A"f 3+1 

It  may  be  remarked  that  the  functions  must  be  com- 
plete in  form  before  beginning  the  operation. 

65.  When  two  integral  functions  of  the  same  variable 
have  a  common  linear  factor,  the  corresponding  equations 
have  a  common  root ;  and  if  the  functions  have  a  com- 
mon quadratic  factor,  the  corresponding  equations  have 
two  common  roots,  etc. 

Ex.  To  find  the  relation  between  the  constants  in  order  that 
the  equations  x^  +  ax -{■  b  =  0  and  x^  +  a^x  +  6i  =  0  may  have  a 
common  root. 

The  functions  x^  +  ax  +  6  and  x^  +  a^x  +  b^  must  have  a  com- 
mon linear  factor. 

A 1+     a+     b      B l  +  «i  +  6i 

bj^A b^  +  b^a  +  b^b      A 1  +  a  +  6 


bB 6  +  &«!  +  6i& 


(a-aO  +  (6-6J 


A',  .  .  .    (b-b,)-\-(ba,-bia) 

These  results  must  be  multiples  of  the  same  Unear  factor. 
Hence  reducing  the  first  term  to  1  in  each  gives  — 


1  A.  ^cti  ~  ^l<^  —  1  _L  ^ 


&i 


b  —  bi  a  —  a^ 

.'.  (b  —  b^y  =  (a  —  a{)(bai  —  b^d); 
which  is  the  required  relation. 


68  LEAST   COMMON   MULTIPLE. 

EXERCISE  IV.  a. 

1.   Find  the  H.  C.  F.  of  each  of  the  following  — 
i.    a:*  -  4  ^3  +  2  a;2  +  4  X  +  1  and  x*  -  6  x^  +  1. 
li.    a^  _  2  a^  +  6  a  -  9  and  3  a*  -  2  a3  -  8 1;;2  +  6  a  -  3. 
Hi.    lOx^  +  x2  -  9ic  +  24  and  20z*  -  11  x-  +  48x  -  3. 
iv.    5ic-^(12a:3  +  4x2  + 7x-3)  and  10x(24x^  -  52a:2  +  14x  -  1). 
V.    ic*  —  px^  +  g  —  1  •  x'^  -\-  px  —  q  and  «*  —  gx^  +  j?  —  1  •  x^  +  gx  —  |). 

vi. 


«_Vl  +  ^and^^^^ 


6*  «*  63      ^3 

2.  Find  the  relation  between  a  and  h  when  x^  _|.  ^^j  +  10  =  0 
and  x2  +  6x  —  10  =  0  have  a  common  root. 

3.  Find  the  value  of  c  when  x^  —  3  x  +  2  and  x^  +  ex  +  3  have 
a  common  linear  factor. 

4.  Find  the  condition  that  ax^  +  6X  +  c  and  jpx^  +  qx  +  r  may- 
have  a  common  linear  factor. 

6.   Find  the  condition  that 

x3  +  ax2  +  6x  4-  0  =  0 
and  x^  +  ajX^  +  ft^x  +  c^  =  0 

may  have  a  common  root. 

6.  Find  the  value  of  a  when  x^  —  x  —  6  and  x^  +  x(3  —  a)  —3  a 
have  a  common  linear  factor. 

7.  If  X*  +  X  +  «  and  x*  —  x  +  6  have  a  common  linear  factor, 
show  that  (a  -  6)*  =  -  8(6  +  a). 

66.  An  expression  which  contains  two  or  more  given 
expressions  as  factors  is  a  common  multiple  of  the  given 
expressions,  and  that  common  multiple  which  is  of  the 
lowest  possible  dimensions  is  the  lowest  common  multiple 
or  the  least  common  multiple  of  the  given  expressions,  the 
Letter  term  being  more  particularly  applicable  to  num- 
bers.    The  contraction  L.  C.  M.  is  used  for  either. 


LEAST   COMMON   MULTIPLE.  69 

If  acef  and  dbde  be  two  expressions  in  which  the 
individual  letters  represent  linear  factors,  their  L.  C.  M. 
is  dbcdefj  and  we  see  that  in  order  to  find  the  L.  0.  M. 
of  two  expressions  or  quantities  we  take  the  factors  that 
are  common  to  both,  as  a  and  e,  and  the  factors  which 
are  peculiar  to  each,  as  c  and  /  from  the  first,  and  b  and 
d  from  the  second,  and  form  the  continued  product  of 
all  these  factors. 

Evidently  a  similar  process  applies  to  the  case  of 
more  than  two  expressions. 

Ex.  1.  To  find  the  L.C.M.  of  x'^-x{a^h)-{-ah,  x^-ax-x-\-a, 
and  x"2  —  6x  —  jc  +  h. 

The  expressions  factored  become  — 

(x  -a)(x-  &),  (x  -  a){x  —  V),  and  (x  -  6) (x  -  1)  ; 

and  the  L.  C.  M.  is  (x  —  a)(x  —  b)(x  —  1);  and  this  is  evenly  divisi- 
ble by  each  of  the  given  expressions. 

Ex.2.    To  find  the  i.  C  itf.  of 

x^  —  ax  —  bx  +  ab  and  x"^  —  2ax  +  a\ 

The  expressions  factored  are  (x  —  a)(x  —  b)  and  (x—a)(x—a). 
Here  sc  —  a  is  a  common  factor,  x  —  b  is  peculiar  to  the  first, 
and  the  second  ic  —  a  to  the  second. 
.-.  the  L.  a  M.  is  (x  -  a)\x  >-  5). 

Ex.  3.    ro  find  the  L.  C.  M.  of 

(x-y-{-z)(x-^y-z),  {y-z-{-x){y-\-z-x),  and  {z-x-^y){z-\-x-y), 
or  a;2  -  (2/  -  zy,  y'^-{z-  x)2,  and  z'^-ix-  y)\ 

The  L.  CM.  is  (x  —  y  +  z)(y  —  z  +  x) (z  —  x  -\-  y\ 

or  —  Sx^  +  'Lx-y  —  2  xyz. 


70  G.  C.  M.   AND   L.  C.  M. 

67.  Theorem.  The  product  of  any  two  quantities  or 
expressions  is  equal  to  the  product  of  their  H.  C.  F,  and 
their  L.  0.  M. 

Let  A  and  B  denote  the  expressions,  and  let  /  be  their 
H.  C.  F.  Then  A=zpf  and  B  =  qf,  where  p  and  q  are  the 
factors  peculiar  to  A  and  to  B  respectively.  Their 
L.  C.  M.  is  pqf. 

But     A  .  B  =pqf=pqf'f=  L,  (7.  M.  x  H,  C.  F, 

Hence  we  may  find  the  L.  C.  M.  of  two  expressions  by 
dividing  their  product  by  their  H.  C.  F.,  and  conversely 
we  may  find  the  H.  C.  F.  of  two  expressions  by  dividing 
their  product  by  their  L.  0.  M. 


APPLICATION    TO    NUMBERS. 

68.  The  foregoing  principles  apply  to  integral  num- 
bers in  the  same  manner  as  to  algebraic  expressions. 

The  H.  a  F,  or,  as  it  is  here  called,  the  G.  G.  M.  of 
the  numbers,  is  the  greatest  number  which  divides  each 
evenly. 

Ex.  1.  To  find  the  G,  0.  M.  of  3824  and  4160.  The 
difference  between  any  multiples  of  the  two  numbers 
must  contain  their  G,  G.  M.  (Art.  ^2), 

Hence,  from  4160  subtract  the  greatest  possible  mul- 
tiple of  3824,  which  in  this  case  is  the  number  itself, 
and  we  have  336.  We  have  now  to  find  the  G.  G.  M.  of 
336  and  3824.  Taking  11  times  336  from  3824  leaves 
128,  and  we  are  to  find  the  G.  G,  M,  of  128  and  336.  By 
continuing  this  process  we  finally  arrive  at  the  factor 
required. 


G.  C.  M.   AND   L.  C.  M. 


71 


The  whole  operation  appears  as  follows : 


A 

.  3824 

B 

.  4160 

A  -11J3'  . 

.     128  . 

.  A' 

B    -   A 

.     336  . 

.  B' 

A'  -     B"  . 

.       48  . 

.  A" 

B'  ~2A' 

.      80  . 

.  B" 

A"-     B"'. 

.      16  . 

.  A'" 

B"  -   A" 
B'"—   A'" 

.       32  . 
.      16 

.  B'" 

The  identical  results  show  that  16  is  the  G,  C.  M. 

This  process  may  be  much  shortened  by  leaving  out 
the  letters  of  reference,  and  by  writing  only  remainders 
in  the  operation. 

Ex.  2.   To  find  the  G.  G.  M.  of  10395  and  20592. 


quotients. 


And  99  is  the  G.  G.  M. 

Explanation.  —  20592  -^  10395  gives  quotient  1  and  remainder 
10197.  10395  H-  10197  gives  quotient  1  and  remainder  198.  10197 
-7- 198  gives  quotient  51  and  remainder  99.  And  lastly  198  -^  99 
gives  quotient  1  and  remainder  99,  or  it  is  a  case  of  even  division. 
Hence  99  is  the  common  factor,  and  hence  the  G,  G.  M. 

The  quotients  are  important  in  the  subject  of  continued 
fractions. 


10395 

20592 

1   >! 

198 

10197 

1 

99 

99 

51 
1    - 

69.  Two  numbers  whose  G.  C.  M.  is  1  are  prime  to  one 
another ;  and  a  number  which  is  prime  to  every  number, 
except  unity,  smaller  than  itself  is  a  prime  number,  or 
simply  a  prime. 

The  following  are  the  primes  less  than  100,  and  a  table 
of  all  primes  below  1000  is  given  at  the  end  of  this  work. 


72  G.C.  M.   AND  L.  C.  M. 

2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  31,  37,  41,  43,  47, 
53,  59,  61,  67,  71,  73,  79,  83,  89,  97. 

All  numbers  not  primes  are  composite  numbers. 

Every  composite  number  can  be  exhibited  as  the 
product  of  prime  factors.  This  is  called  the  composition 
of  the  number. 

Let  ^denote  a  composite  number. 

Divide  JSf  by  2,  and  the  resulting  quotient  by  2,  and  so 
on  until  a  quotient  is  found  which  is  not  divisible  by  2. 
Call  this  quotient  N'.  Divide  JV  by  3,  and  the  quotient 
by  3,  and  so  on,  until  a  quotient,  N",  is  found  which  is 
not  divisible  by  3.  Divide  N^'  by  5,  etc.,  and  continue 
the  operation  by  7, 11,  etc.,  using  only  primes  as  divisors. 

Then  if  2  has  been  used  a  times  as  a  divisor,  3  b  times, 
5  c  times,  etc., 

JV^=2«.3^.5^... 

And  this  is  the  composition  of  the  number  N. 

Ex.  1.  The  composition  of  8640  is  2^ .  33 .  6  j  and  8640  is  said  to 
be  decomposed  into  its  prime  factors. 

Ex.  2.    To  find  the  G.  O.  M.  of  8640  and  1720. 

Composition  of  8640  is  2^  .  S^  •  5. 
Composition  of  1720  is  2^ .  5  •  43. 
And  the  G,  C.  M.  is  23 .  5  =  40. 

70.  To  find  the  L.  (7.  M.  of  two  or  more  numbers,  we 
may  decompose  the  numbers  into  their  prime  factors, 
and  take  the  highest  power  of  each  factor  involved,  and 
form  their  continued  product. 

Ex.    To  find  the  L.  C.  M.  of  8640,  1280,  and  1560. 

8640  =  26.33.5,   1280  =  28.5,   1560  =  23.3.5-13. 
And  the  L.  C.  M.  is  28 .  33 .  5  •  13,  or  449280. 


G.C.M.   AND   L.C.M.  73 

The  operation  may  also  be  conveniently  carried  out  as 
follows : 

40  8640  1280  1560  40  is  the  G.  C.  M.  of  the  three  numbers. 
8  216  32  39  8  is  the  G.  C.  M,  of  216  and  32,  and  is 
3        27  4        ...  prime  to  39. 

9        ...        13        3  is  the  G.  G.  M,  of  27  and  39,  and  is 
prime  to  4. 

The  quotients  9,  4,  13  are  prime  to  each  other. 

Hence  40  x  8  x  3  x  9  x  4  x  13,  or  449280  is  the  L.  C.  M. 

EXERCISE  IV.  b. 

1.  Find  the  i.  CM,  of  the  following  — 
i.    12x  -  36,  x2  -  9,  x2  -  5«  +  6. 

ii.   ic2  -  (a  +  h)x  +  ah,  aj^  -  (6  +  c)x  +  &c,  x'^—{c-\-  a)x  +  ca. 
iii.   1  -\-p+p\  1  -p  +i52,  1  +^2  4.^4. 

2.  If  ax"^  +  hx-\-  c  and   cx^  +  6x  +  a   have  a  linear  common 
factor,  show  that  a  ±h  -\-  c  =  0, 

3.  Find  the  compositions  of  the  numbers  — 

i.  72.        ii.  180.        iii.  824.        iv.  1048.        v.  25200. 

4.  Find  the  G.  C.  M.  of  144,  840,  5040. 
6.  Find  the  L.  C.  M.  of  the  9  digits. 

6.  What  is  the  least  multiplier  that  will  make  720  a  complete 
square  ?    That  will  make  1440  a  complete  cube  ? 

7.  What  is  the  lowest  factor  that  will  make  a^  —  45c2-f-5x  —  2 
a  complete  square  ? 

8.  What  is  the  least  multiplier  that  will  make  144  a  multiple  of 
64  ?    That  will  make  x3-5x2  +  5x-la  multiple  of  x^  -  4a;  +  3  ? 


CHAPTER  V. 

Fractions.  —  Symbols  qo  and  0. 

71.  The  expression  3/4  denotes  that  3  is  to  be  divided 
by  4.  As  numbers,  3  cannot  be  divided  by  4 ;  and  hence 
we  indicate  this  impossible  arithmetical  operation  sym- 
bolically by  writing  it  in  a  form  of  division,  and  we  call 
the  whole  symbol  a  fraction.  We  then  discover  the  laws 
of  transformation  of  these  symbols,  and  these  laws  form 
the  working  rules  for  fractions. 

In  this  relation  the  dividend  is  called  the  numerator 
and  the  divisor  the  denominator. 

The  expression  a/h  is  an  algebraic  fraction,  in  which 
a  and  h  stand  for  any  numbers  or  expressions ;  and  the 
transformations  of  this  symbol  must  apply  to  arithmetical 
fractions  as  particular  cases. 

RULES    OF   TRANSFORMATION. 

72.  The  fractional  form  ^  +  ^  +  ^+-'^  ^-^^^  ^^^  Ylhq 

written  beneath  a-\-h-\-c-\ (Art.    10),   shows   that 

a-\-h-{-c-\ is  to  be  taken  in  its  totality  to  form  the 

numerator. 

(1)  Let  Q  =     T"   ,  Q  denoting  the  fraction  as  a  whole. 

Then  QD  =  a  +  &  (Art.  37)  ;  and  if  we  take  d  such  a 

multiplier  that  Dd  =  1,  and  therefore  d  =  — ,  we  have 

74 


FRACTIONS.  75 

QDd  =  ad-{-bd,  or  Q  =  ^-\.^. 

Hence  the  denominator  of  a  fraction  is  distributive 
throughout  the  terms  of  the  numerator. 

And  conversely,  the  algebraic  sum  of  any  number  of 
fractions  with  the  same  denominator  is  that  fraction 
whose  numerator  is  the  algebraic  sum  of  the  numerators, 
and  whose  denominator  is  the  same  denominator. 

(2)  Let  Q  =  -.  Then  QD  =  ]Sr;  and  p  being  any 
multiplier, 

QDp  =  Np. 

...  Q  =  ^=^. 
^     Dp     D 

Hence,  multiplying  both  numerator  and  denominator 
by  the  same  multiplier  does  not  alter  the  value  of  the 
fraction. 

(3)  For  p  write  -,  and  we  have  from  (2) 


Q  = 

N 

Q 
'  D' 

1 

~d' 

And  hence 

dividing  both  numerator  and  denominator 

by   the   same 

divisor 

does  not  alter  the  value 

of 

the 

fraction. 

(4)  Let 

Q  = 

W 

'd' 

and 

^      D' 

Then 

and 

r.  an 

Q'D'  =  N'. 

DD' 


76  FRACTIONS. 

Hence,  to  multiply  two  fractions  together,  we  multiply 
together  the  numerators  for  a  new  numerator  and  the 
denominators  for  a  new  denominator. 

(5)  From  (4),  -^ — -  =  — ^,  and  multiplying  both  frac- 

tions  by  -—  gives  -^  =  —  •  — -  • 

Hence,  to  divide  one  fraction  by  another  we  multiply 
the  dividend  by  the  inverted  form  of  the  divisor. 


(6)   Let  Q  =  — ;  and  since  p  =  ^, 


P      P 

Therefore,  we  multiply  a  fraction  by  a  given  quantity 
when  we  multiply  the  numerator  by  that  quantity,  or 
when  we  divide  the  denominator  by  that  quantity. 

Also,  by  writing  -  for  p,  we  see  that  we  divide  a  frac- 
tion by  a  given  quantity  when  we  divide  the  numerator 
by  that  quantity,  or  when  we  multiply  the  denominator 
by  that  quantity. 

Co7\  If  two  fractions  have  equal  denominators  the 
greater  fraction  has  the  greater  numerator,  and  con- 
versely. 

And,  if  two  fractions  have  equal  numerators,  the 
greater  fraction  has  the  smaller  denominator,  and  con- 
versely. 


FRACTIONS.  77 

73.  The  operations  with  fractions  are  as  general  in 
character  as  those  with  whole  numbers,  and  beyond  the 
rules  of  transformation  as  now  established  no  general 
directions  can  be  laid  down.  The  principal  operation  on 
fractions  in  themselves  is  the  simplification  of  their 
forms.     Facility  in  this  is  the  result  of  practice. 

Ex   1     3a;2  -f  x  -  2_  (3a;  -  2)(x  +  1)  _3a;  -  2 
2x^-x-S~(2x-S)(_x+r)~2x-S 

X  -f  1    .    X  —  1 

w  2     a;  -  1     X  -{-  l_(x  +  ly  +(x  -  iy_x'^  +  1 

'  x  + 1    X - 1~ (X -{■  ly -(x - ly^-  2x 

x-l     x-\-l 

Ex  3     a^^-1.     x  +  1     _(x^-l)(Sx^  +  2x) 
»2+l'3xH2x        (a:2  +  l)(x+l) 

=  Sx^-Sx^  +  2x-  a^(5x2-a;  +  4)^ 

(X2  +  1)(X  +  1) 

74.  From  Art.  72  (1),  to  add  fractions  we  bring  them 
to  have  the  same  denominator.  We  then  add  the  numer- 
ators, and  place  the  sum  over  the  common  denominator. 

Subtraction  being  addition  with  changed  signs  follows 
the  same  rule. 

When  several  fractions  are  to  be  added  the  L.  C.  M.  of 
all  the  denominators  is  the  simplest  common  denominator. 

Ex.  1.   Simplify 

a  ,  h  ,  c 


{a  —  6)  (a  —  c)      (h  —  c)(b  —  a)      (c  —  a){c  —  b) 

The   L.  C.  M.    of  the  denominators  is  (a  —  h){h  —  c)(c  —  a)  ; 
and  the  numerators  become  respectively  — 

«(c  —  &),  b(a  —  c),  and  c(b  —  a). 


78  FRACTIONS. 

And  the  sum  is  g(c^i&I+A(^' -  «)+ Ha  -  c). 
(a  —  b)(b  —  c){c  —  a) 

And  as  the   numerator  vanishes   upon  distribution,    and  the 
denominator  does  not,  the  sum  of  the  three  fractions  is  zero. 


Ex.  2.    Simplify 
a 


+  -7-. ..,  .. 7T+- 


(a— 6)(a— c)(ic— a)      (b  —  c){b  —  a){x—b)      {c—a)(c  —  b){x  —  c) 
The  L.  C.  M.  of  the  denominators  is 

(a  —  b){b  —  c) (c  —  a)(x  —  a) (x  ~-  b)(x  —  c); 
and  the  sum  of  the  new  numerators  is 

—  a(b  —  c)  {x  —  b)  (x  —  c)  —  b{c  —  a)(x  —  c)(x  —  a) 
—  c(a  —  b)(x  —  a)(x  —  b). 
This  latter  expression  factors  into 

x(a  —  b){b  —  c)(c  —  a). 
Therefore  the  simplified  sum  of  the  fractions  is 

x 

(x  —  a){x  —  6)  (x  —  c) 

EXERCISE  V.  a. 

1.    Simplify  the  following  — 

.    x'^-h  jxy  +  y^^ 
ic2  +  I  xy  -  2/2 


\x    a/\x    aj\      x-{-aJ 


.J    3x-2+  lOxy +  3y2  ^        gg  +  &^  +  c^  -  3 abc 

*    3x2 +  8x2/ -3 2/2*  *   a'^+b'^-^c^-ab-bc-ca 


m. 


w(x2  —  2/2)  +  (?i2  —  l)x2/  *  \l+x       X    y  "  \1  +  X       X    J 

^^^  a  +  c &  +  c x  +  c 

(a  —  6)  (x  —  a)      {a  —  b)(x  —  b)      (x  —  a)  (x  —  b) 

iii.  — ?^ + t + ^ 

ix-y)(x-z)      {y-x){y-z)      (z-x){z-tj) 


FKACTIONS.  79 

ix.  s| ^ 1.        xi.  s| — y^^ ). 

(  1  )  xii  s|i3L±i^HL±Ml. 

xiii.  2  \ ^^^  "^  ^^ —  I ,  where  x  is  not  varied. 

<>(«  —  5) (a  —  c)  J 

I  (a  -  5)(a  -  c)  i  I  («  _  5)((i  _  c)  / 

2.  If  a  =  -^— ,  &  =  -^— ,  c  =  -^-,  show  that 

2/  +  2  z-\-x  x  +  y 

ic2  ?/2  2;2 


a(l  —  6c)      &(1  —  ca)      c(l  —  a6) 

8.  Find  the  value  of  ^  +  V  - '^ 

x-y  +  1 

when  aj  =  -^±A.  and  y  =  ^^±^^ 

«6  +  1  a6  +  1 

*i2        /y2        r2 

«2  ^  62  ^  C2 

a2  52  g2  ^-2       ^2         (.2 

6.   If  1+^  =»  .  1  +  ^  +  ^',  show  that  x^  =  ^^^. 
1— xal  —  x  +  ic2  5  +  a 

6.  Simplify  j>+I£-  «U  1 1  ^fl±I2 .  ?!  when  .2  +  ^2=1. 
l/c  —  rs     c  i       V        fc  —  rs    c) 

75.    Let  ic  = ,  where  A,  a,  b  are  all  positive  finite 

a  —  b 

quantities  and  b  is  not  greater  than  a. 

If  &  is  <  a,  X  has  some  positive  finite  value,  and  the 
nearer  b  approaches  to  a  in  value  the  greater  the  value 


80  FRACTIONS. 

of  X  becomes.  By  making  the  difference  between  a  and 
h  small  enough  we  can  make  the  value  of  x  as  great  as 
we  please. 

Thus,  let  A  be  10.  If  a  —  &  is  1,  a;  is  10 ;  if  a  —  h  is 
0.1,  a;  is  100 ;  \ia-h  is  0.00001,  x  is  1000000;  etc. 

When  a  —  h\B>  made  smaller  than  any  conceivable  quan- 
tity, X  becomes  greater  than  any  conceivable  quantity. 

In  this  case  h  is  said  to  approach  infinitely  near  to  a 
in  value ;  the  difference  between  a  and  h  is  then  denoted 
by  0,  and  the  value  of  x  is  denoted  by  oo,  read  infinity. 

Thus  we  say  that  ^  ^  ^  ^  any  finite  quantity^    ^^^ 
•^0  0 

assuming  that  these  symbols  obey  the  formal  laws  of 
quantitative  symbols, 

00  X  0  =  any  finite  quantity. 
But  1  is  a  finite  quantity,  and  x  =  QO  ; 
therefore         -  X  0,   or   -  =  any  finite  quantity. 

The  expressions  cx)  x  0  and  -  are  mere  symbols  having 

no  particular  value  except  through  their  history ;  that  is, 
through  a  knowledge  of  the  source  whence  they  have 
come.     The  expression  -,  however,  which  occurs  quite 

often,  does  not  necessarily  mean  zero,  but  may  mean  any 
finite  quantity  whatever. 

Also,   since  oo  x  0  =  ^4,  0  =  — ,  and  hence  any   finite 

00 

quantity  divided  by  infinity  gives  zero  as  a  quotient. 

Moreover,  we  are  not  justified  in  writing  go  —  go  =  0, 
or  00  ^  00  =  1,  for  oo  does  not  mean  any  definite  quan- 


FRACTIONS.  81 

tity,  but  merely  a  quantity  so  great  as  to  be  undefinable 
and  inconceivable. 

76.  Special  Roots.  (1)  When  an  integral  function 
contains  the  variable  as  a  monomial  factor  in  the  first 
degree,  one  of  the  roots  is  zero. 

Thus       ^  +  3ax^ -\-2a^x  =  0  =  x{x  +  2a)  {x-{-a); 

which  is  satisfied  by  a;  =  0,  since  the  whole  then  vanishes. 
The  roots  are  accordingly,  0,   —  2  a,   —  a. 

If  the  variable  is  of  two  dimensions  in  the  monomial 
factor,  two  of  the  roots  are  zero. 

(2)  In  the  linear  equation  x  =  x-\-  a,  to  which  equa- 
tions are  sometimes  reducible,  we  transfer  x,  and  obtain 
x  —  x  =  a. 

Now  we  are  not  justified  in  saying  that  a:  —  a;  =  0,  and 
therefore  a  =  0,  for  a  is  a  given  quantity  whose  value  is 
not  at  our  disposal ;  we  must  endeavor  to  find  some  value 
for  x  that  will  satisfy  the  equation. 

But  X  — x  =  x(l -'l)  =  a. 

a 


1-1 


=  oo  by  Art.  75. 


Whence  x  =  ck>  is  a  symbolic  root  of  the  equation. 
The  meaning  of  the  solution  is  that  a  being  finite,  the 
larger  x  is,  the  more  nearly  is  the  equation  x  =  x-^  a 
satisfied,  but  that  it  cannot  be  completely  satisfied  by 
any  finite  value  of  x.  We  shall  return  to  this  again  in 
Art.  81. 

77.  Let  x^-{-ax-{-b^x^  -{-px  +  q,  where  a,  b,  p,  q  are 
^11  finite  quantities. 


82  FRACTIONS. 

If  X  is  finite,  a^  on  one  side  cancels  o^  on  the  other, 
and  we  have  a  single  finite  value  of  x,  namely, 

q-b 
a—p 

(1)  If  q  =  b,  and  a=p,  a;  =  -  =  any  finite  quantity 
by  Art.  75,  and  the  solution  is  indefinite, 

(2)  If  g  —  6  is  not  zero,  and  a  =  p,  x=  x>, 

(3)  li  q  =  h  and  a  —p  is  not  zero,  a;  =  0. 

Again,  as  the  equation  is  quadratic  it  must  have  two 
roots.  Art.  59  (2). 

Dividing  throughout  by  x^  gives 

X       X^  X       x^ 

and  the  larger  x  becomes  the  more  nearly  does  this 
become  an  identity. 

Hence  ic  =  oo  is  a  solution. 

As  the  equation  may  be  written 

we  infer  that  if  the  coefficient  of  the  square  term,  in 
a  quadratic  equation,  becomes  zero,  one  solution  of  the 
equation  is  a;  =  oo. 

Or  more  generally,  if,  in  any  integral  equation,  the 
coefficient  of  the  highest  power  of  the  variable  becomes 
zero,  one  root  of  the  equation  is  oo. 


FRACTIONS.  83 


EXERCISE  V.  b. 

1.  Given  ?^+_§  +  ^  ^  3x  +  4,  to  find  x. 

6  3 

2.  Given  ^-±^  +  ?^  =  5,  to  find  a;. 

x  +  2     3a;     3 

3.  Given 

a  c  ,  X  1 


(x  —  a)(x  —  c)      (a  —  c)  (a  —  x)      (c  —  a)(c  —  x)      a  —  c 
to  find  X.     What  is  the  value  of  x  v^rhen  c  =  0  ?  when  a  =  0  ? 

4.  Is    /'l+?Wl  +  ^'\  =  /'l+iyi  +  ^'\    an  equation  or  an 
\        dj  \x     xj      \        d J  \        aj / 

identity  ?    What  value  has  x  ? 

6.   What  expression  substituted  for  x  will  make 

Zx^-l      x±2^  =  1  an  identity  ? 
a-l       a  +  2  -^ 

6.  From   _^— +— ^ 2(a;2  -  a6)      ^  ^^  ^^^    ^  general 

X  —  a     X  —  &      {x  —  a){x  —  h) 
value  of  X  ;  and  also  the  particular  value  when  a  +  &  =  0. 

7.  Given  (^-"^>^^  +  ^)  =  ^il,  to  find  x. 

(x-2)(x+l)      x-1 

8.  Given  C^^  -  1)  (^  +  2)  (a;  -  3)  ^  i  ^  q,  to  find  all  the  values 

(x  +  l)(x-2)(x  +  3) 
of  «. 

9.  Given  ^x^-x^-Qx-^2    4:X^  +  x^-^x-2  2       ^^^ 

X  —  1  X  ic(l— x) 

to  find  all  the  values  of  x. 

10.  If  ^  +  ?  =  ^  +  ^,  what  relation  holds  between  a  and  &  ? 

X     b     X     a 

11.  Two  numbers  differ  by  10,  and  one-half  the  less  is  greater 
by  1  than  one-sixth  the  greater.     Find  the  numbers. 


84  FRACTIONS. 

12.  One  body  moves  about  a  circuit  in  a  days,  and  another  in 
h  days,  and  they  start  from  the  same  point.  How  many  days  will 
elapse  between  two  conjunctions  ? 

13.  The  sun  moves  in  the  ecliptic  0°.9856  per  day,  and  the  moon 
moves  13°.  1690  per  day.  Find  the  days  elapsing  between  two  new 
moons. 

14.  Find  a  number  such  that  if  a  be  added  to  it  and  h  be  sub- 
tracted from  it,  the  difference  of  the  squares  of  the  results  shall  be 
the  number. 

What  relation  must  hold  between  a  and  h  that  the  number  may 
be  (1)  zero,  (2)  infinity  ? 

15.  Given  ^Ji3  _  ^+_4  ^-_6  ^  x'^-2x-Vo  ^^  ^^^  ^ 

x+\      5C+2      ic-4  x2-9 

16.  Given  -^-  +  ^^  =  ^-±^  +  ^^,  to  find  all  the  values 

X  —  2     X  —  1     X  —  1     X  —  Q 
of  X. 

17.  Given         a  +  x  a-x  3a  _    ^ 

cC^  -\-  ax-\-  x^     a^  —  ax-\-  x^     x{a^  +  d^x^  +  x*) 
find  X. 

18.  Divide  $64  among  A,  B,  and  C,  so  that  A  may  have  3  times 
as  much  as  B,  and  C  have  I  as  much  as  A  and  B  together. 

19.  A  person  spends  .|2  and  then  borrows  as  much  as  he  has 
left.  He  again  spends  $2  and  borrows  as  much  as  he  has  left ;  etc. 
After  his  fourth  spending  he  has  nothing  left.  How  much  had  he 
at  first  ? 

20.  A  person  spends  $a,  and  borrows  as  much  as  he  has  left ; 
then  spends  $a,  and  borrows  as  much  as  he  has  left,  etc.,  for  n 
times,  when  he  has  nothing  left. 

2n  1 

Show  that  he  had  at  first  a  dollars. 

2n-l 

21.  In  a  naval  battle  the  number  of  ships  taken  was  7  more,  and 
the  number  burnt  2  less,  than  the  number  sunk;  15  escaped,  and 
the  fleet  consisted  of  8  times  the  number  sunk.  How  many  ships 
were  in  the  fleet  ? 


FBACTIOKS.  .  85 

78.   The  following  properties  of  equal  fractions  are  of 
special  importance : 

I.   Let  ^  =  |, 

where   a,   b,  c,  d  denote  any  quantities  or  expressions 
satisfying  the  indicated  relation. 

(1)  Multiplying  by  -  gives  -  =  -. 

C  C        U/ 


(2)  Adding  1  to  each  member 

(3)  Subtracting  1  from  each  member 

(4)  Dividing  (2)  by  (3) 


a-\-b  _c-\-  d 

h     "     d    ' 

a—b      c—d 


b  d 

a  +  b  _c-}-d 
a  —  b      c  —  d' 


Eelations  (1),  (2),  (3),  (4)  are  all  direct  consequences 
of  the  original  relations. 

The  number  of  such  derived  relations  is  unlimited; 
those  given  are  of  most  importance. 

It  will  be  noticed  that  these  expressions  have  that 
kind  of  correlative  symmetry  by  which  we  may  inter- 
change a  and  b  if  we  interchange  c  and  d,  or  we  may 
interchange  a  and  c  if  we  interchange  b  and  d,  but  in 
general  we  cannot  interchange  a  with  c  or  6  with  d. 

11.    Let  ^  =  ^  =  ?  =  ...  =  Q,  say. 

Then  a  =  bQ,  c=:dQ,  e=fQ,  etc. 

And         Q(lb  -{-md-\'nf-\ )  =  la-{-mC'{'ne+  "• 

rK\  n      ^  Za +  mc  4-ne -f- ••• 


b  lb  -i-  md  -\-  nf-\-  < 


86  •  FRACTIONS. 

By  giving  particular  values  to  Z,  m,  n  •••  an  indefinite 
number  of  special  relations  may  be  obtained. 

The  relations  of  I.  and  II.  are  frequently  employed 
with  great  advantage,  the  letters  a,  h,  c,  etc.,  being 
general  symbols  denoting  any  quantities  or  algebraic 
expressions. 

^2 


Ex.  1.    To  find  X  from  the  equation  f^il^V 

\a  —  xj 

cfi  -\- 2  ax  +  x"^  _  ah  -^  ex 


ah 


Expanding,  etc., 
Applying  (4)  of  I., 


■2ax-\-  x'^  ah 

a?'  -\-  x^  _2  ah  +  ex 


2  ax  ex 

Subtracting  denominator  from  numerator  for  a  new  numerator, 

(ia-xy^2ah 
2  ax  ex 

.'.  ex(a  —  xY  =  ^  a'^hx. 
Whence         x  =  0,  (76.1),  and  x  =  a-2aJl-Y 

Ex.  2.  If  ^  =.  ^ ,  then  (^'^  +  ?>-^)  («  +  6)  ^  q^, 
b     d  (c^  +  d')(ic-{-d)      c3 

Exercises  of  this  kind  may  be  solved  in  several  ways  :  as  (1)  by 
transforming  one  expression  into  the  other  ;  (2)  by  using  the  first 
relation  to  show  that  the  second  is  an  identity,  etc. 

m  ^  =  ^.      .    Q^^+  ^^  ^  c2  +  d^    ^^^  a^+  &2  ^b^^c^ 

^    ^    62        ^2*        •  •  5-2  cf2      '  C2  +  d^        d^        C2' 

by  relations  (1)  and  (2),  I. 

Again,  «+^=L±i,  and«+^  =  ^  =  «.    ■ 

h  d  e  +  d     d      c 

.:  by  multiplication,   «^±^  •  "L^^  =  «!  .  ^  =  «!.  q.  k.  d. 


FRACTIONS.  87 


(2)  Let  -  =  -  =p;  then  a  =  bpy  c  =  dp, 
h      d 

Substitute  for  a  and  c  in  the  second  expression,  and  it  becomes 
hHp'+lXp-\-^)  ^  m^  an  identity. 

Ex.  3.   If  — ^—  =  — ^—  =  — - — ,  each  fraction  =  1. 

2y  —  z     2z  —  X     2x  —  y 

By  (5),        ^      - x  +  y-\-z a;  +  .V  +  g  _.  2. 

'  2y  —  z      2y  —  z  +  2z  —  x-\-2x  —  y     x -\-  y  +  z 

Ex.4.    If    ^  =  ^  =  ^^^    and    £^+^  +  ?!=l,   to  find    the 
X         y         z  a^     h'^     c^ 

values  of  x,  y,  and  z  in  terms  of  the  remaining  letters. 
The  first  relation  gives    ^^'^^21; 

^     y      1 

a       b        c 
whence,  squaring  each  fraction  and  employing  (5), 

x2  1 

an 


Whence  x  = 


^{aH^-h  &2m2+  c2n2) ' 
with  symmetrical  expressions  for  y  and  z. 

Question  4  furnishes  an  example  of  collateral  symmetry 
between  the  two  sets  of  letters  a,  b,  c  and  l,  m,  n.  Thus 
if  we  change  a  to  &  and  b  to  c,  we  must,  at  the  same  time, 
change  Z  to  m  and  m  to  n.  But  we  are  never  supposed 
to  make  an  interchange  between  letters  from  different 
sets.  In  like  manner  we  may  have  collateral  symmetry 
amongst  three  or  even  more  sets  of  different  letters. 


88  FRACTIONS. 


EXERCISE  V.  C. 

1.  If  ^=:^  =  ^=...     Show  — 

h      d     f 

.     (a-c)2  +  (6-(?)2^g2  4.;^2 
'(a-c)2-(6-(?)2     a2-62* 

ii    g^  4-  g^ft  +  q&^  ■\-h^  ^  b^ 

...     g  _  V(Z2g2  +  w2c2  +  nV) 
6      VG^^'^  +  ^'^^^^  +  ^T-^)* 

2.  If  ^  =  ^  and  ^  =  ^,  then  «V^  +  5V^^  gy^- fty^, 

8.   If  ^  =  ^  =  ^,  (a2  +  52+c2)(62  +  c2  +  c?2)  =  (a6  +  &c+c(Z)2. 
0      c     d 

4.   Under  the  conditions  of  Ex.  3,  show  that  ' 

V(g&)  +  V(&c)  +  y/iicd)  =  V{(g  +  &  +  c)  (6  +  c  +  d)}. 

6.   If  ^        =^ 


then  ?(y  -  2j)+  |(5?-a)+  £(x  -y)=0. 


6.    Given  that  ga;2  +  5^2  +  2  ^  =  0,  and  ^  =  ^  =  ^  =  ?, 

i       m      n     p 

show  that  -  +  —  +  2  w»  =  0. 

a       6 


7.  If    ^^-y^   ^    y^-zx  ^  ^^^^  fraction  =  x  +  y  +  z. 

x(l  —  2/^)     y(l  —  ^x) 

8.  IfgLLJ^     ^  +  g    =    g  +  «    ,8g  +  96  +  5c  =  0. 

a- 6      2(6 -c)      3(c-a) 


FRACTIONS.  89 

9.   K  f±^  =  ^-±^  =  ^±1,  then  each  fraction  is 
0  —  c     c  —  a     a  —  h 

If x^  +  y^  +  z^ -» 

A/ \(6-c)2  +  (c-a)2+(a-6)2/* 

10.   If  -:^  =  -^  =  -^=2R,  and  a  +  b  +  c  =  2S, 

smA     smB     sm  (7 

then  >S  =  i2(sin^4-sin^+ sinC). 


CHAPTER  VI. 

Ratio,  Proportion,  Variation,  or  Generalized 
Proportion. 

79.  The  ratio  of  a  to  &  is  the  quotient  arising  from 
dividing  a  by  h,  where  a  and  h  denote  any  numerical 
quantities.  If  the  division  is  even,  the  ratio  is  an 
integer,  and  is  expressible  ;  if  uneven,  the  ratio  is  a 
fraction  and  can  only  be  indicated. 

In  this  relation  a  and  h  are  called  the  terms  of  the 
ratio,  a  being  the  antecedent  and  &  the  consequent. 

The  ratio  is  commonly  symbolized  as  a :  6. 

If  a  >  &,  the  ratio  is  one  of  greater  inequality. 

If  a  =  b,  it  is  one  of  equality  ;  and  if  a<b,  it  is  one 
of  less  inequality. 

When  two  ratios  are  multiplied  together,  after  the 
manner  of  fractions,  they  are  said  to  be  compounded. 

Thus  ac  :  bd  is  compounded  of  a :  &  and  c :  d. 

When  a  ratio  is  compounded  with  itself,  the  terms 
are  squared,  and  the  result  is  the  duplicate  ratio  of  the 
original.     Thus  a^ :  b^  is  the  duplicate  of  a  :  6. 

Similarly,  a^ :  b^  is  the  triplicate  of  a  :  6 ;  and  -y/a^ :  -y^b^ 
is,  in  physics,  sometimes  called  the  sesquiplicate  ratio  of 
a:  b. 

80.  As  a  ratio  is  virtually  a  fraction,  all  the  laws  of 
transformation  for  fractions  apply  to  ratios. 

The  ratio  of  one  quantity  to  another  does  not  depend 
90 


BATIO.  91 

upon  any  absolute  magnitudes  of  the  quantities  (for 
there  is  no  absolute  magnitude),  but  upon  the  relative 
magnitudes  of  the  quantities. 

It  is  thus  that  the  ratio  of  one  quantity  to  another 
expresses  the  true  relation  of  magnitude  or  greatness 
existing  between  the  quantities. 

The  ratio  - :  -  is  the  same  as  a:b,  whatever  x  may  be. 

X    X 

But  if  X  is  very  small  as  compared  with  a  and  b,  both 
terms  become  very  great ;  and  if  x  is  very  great  as  com- 
pared with  a  and  b,  both  terms  are  very  small ;  but  the 
relation  of  greatness  existing  between  the  terms  remains 
the  same. 

If  x  =  0,  both  -  and  -  become  infinite ;  so  that  oo  :  oo 

X  X 

may  be  any  ratio  whatever. 

If  a;  =  00,  both  -  and  -  becomes  zero ;    so  also   0  :  0 

X  X 

may  be  any  ratio  whatever.     (Compare  Art.  75.) 

81.  Theorem.  The  addition  of  the  same  quantity  to 
both  terms  brings  the  ratio  nearer  to  unity,  or  to  a  ratio 
of  equality. 

Let  aibhe  the  ratio,  and  let  x  be  added  to  each  term, 
making  a-\'X:b  +  x. 

Then  ^      a  +  x_x(a-b)  __^ 

^^^^  b~'bTx=b(b  +  ^'^'     ^' 

and  ^-1^^       =Q,say. 

All  the  letters  denoting  positive  quantities,  if  a  :  6  >  Ij 
a>b  and  Q  and  Qi  are  both  +. 


92  PROPORTION. 

.*.  a  +  xib  +  xKaib,  and  >  1 ; 
or  a-]-x:b'\'X  lies  between  a  :  b  and  1. 

If  a:b <lf  a <b,  and  Q  and  Qi  are  both  — . 

.-.  a-{-x:b  +  x>a:b,  and  <  1 ; 
or  a-^-xib  +  x  lies  between  a  :  b  and  1, 

which  proves  the  theorem. 

Cor,  1.  By  writing  --  x  for  x,  we  see  that  to  subtract 
the  same  quantity  from  both  terms  of  a  ratio  of  inequality 
removes  the  ratio  further  from  1,  provided  the  subtrac- 
tion leaves  both  terms  positive. 

Cor,  2.  Qi  decreases  as  x  increases,  and  by  making  x 
great  enough,  we  may  make  Qi  as  small  as  we  please. 
That  is,  by  adding  the  same  quantity  to  both  terms  of  a 
ratio  we  may  bring  the  ratio  as  near  1  as  we  please. 

We  have  here  another  proof  of  Art.  76,  for  if  x-\-  a  = 
x-\-b,  we  have  x-\-a:  x-{-b  =  lj  and  whatever  values  a 
and  b  may  have,  provided  they  are  finite,  the  statement 
is  satisfied  by  x  —  cc,  since  that  value  for  x  makes  the 
ratio  to  differ  from  1  by  a  quantity  less  than  any  assign- 
able quantity. 

PROPORTION. 

82.  Pour  quantities  are  proportional,  or  are  in  pro- 
portion, or  form  a  proportion,  when  the  ratio  of  the  first 
to  the  second  is  equal  to  the  ratio  of  the  third  to  the 
fourth. 

Thus  a,  b,  c,  d  are  proportional  when 

a:b  =  c:d. 


PROPOKTION.  93 

This  may  be  expressed  as    ^  =  -,  and  a  proportion, 

being  thus  an  equality  of  two  fractions,  is  best  dealt  with 
after  the  manner  of  fractions. 

The  proportion  is  evidently  subject  to  all  the  trans- 
formations of  Art.  78,  I. 

83.  In  the  proportion  a:b  =  c:d,  a  and  d  are  the 
extremes  and  b  and  c  the  means. 

But  since  the  same  proportion  may  be  written  b:  a=: 
d  :  c,  the  extremes  and  means  are  capable  of  exchanging 
places. 

Writing  the  proportion  -  =  -,  or  -  =  -,  or  -  =  -,  etc., 
b     d        c      d        a     c 

which  all  express  the  same  relation,  we  may  represent 
the  form  generally  by  — ! — ,  where  the  letters  are  writ- 
ten in  the  four  corners  formed  by  the  two  crossed  lines. 

In  this  form  a  and  d,  as  also  b  and  c,  are  opposites  of 
the  proportion,  as  standing  in  opposite  corners,  and  we 
can  make  the  general  statement 

The  terms  of  a  proportion  may  be  written  in  any  order, 
provided  the  opposites  are  unchanged. 

By  cross-multiplication  ad  =  be.  That  is,  when  four 
quantities  are  in  proportion,  the  product  of  each  pair  of 
opposites  is  the  same ;  and  conversely,  if  two  equal 
quantities  be  each  divided  into  any  two  factors,  these 
factors  form  a  proportion,  of  which  the  factors  of  the 
same  quantity  are  a  pair  of  opposites. 

84.  If  a :  6  =  c :  c?^  d  is  a  fourth  proportional  to  a,  b, 
and  c. 


94  PROPORTION. 

If  a:b  =  b  :  c,  b  is  Si  mean  proportional,  or  a  geometric 
mean  between  a  and  c.     In  this  case  b  =:^(ac). 

If  a:b=b  :  c=c:  d=etc.,  the  statement  is  a  continued 
proportion. 

Ex.  1.    If  a  +  b  :  a  =  a  —  b  :  b,  to  find  a  :  b. 

Here  ba  -\-  b^  =  a^  —  ab  ;  whence  a  —  b  =  b^2, 

and  a  =  5(1  +^2),  or  a:b  =  l-\-  ^2. 

Ex.  2.  If  ^2^2  _  52^.2  _  ^252^  to  find  approximately  the  ratio 
X :  y  when  x  becomes  indefinitely  great,  and  a  and  b  are  finite 
constants. 

Evidently  y  becomes  indefinitely  great  with  x.  Dividing  by  x^ 
the  relation  gives  a2  .  1^  _  ^2  _  «!^. 

When  ic  approaches  00,  - —  approaches  0,  and  ^  approaches  — • 
x2  x^  a^ 

And  when  x  =  qo,        ^  i=  -j-  ^,  or  x:y  =  ±  a:b. 
y         b 

Ex.  3.   Two  numbers  are  in  the  ratio p:q;  what  must  be  added 
to  each  that  the  ratio  of  the  new  numbers  may  be  P :  Q  ? 
Let  mp,  mq  be  the  numbers,  and  add  x  to  each. 
Then  mp -}■  x:mq-{- x=  P:  Q. 

whence  ^^m(pQ-gP). 

P-Q 

Cor.  When  P:§=1,  P=Q,  and  the  value  of  a  becomes 
infinite. 

Ex.  4.  To  find  4  numbers  in  continued  proportion  such  that 
their  sum  may  be  66. 

Let  a,  6,  c,  d  be  the  numbers. 

Then  ^=^  =  ^=z,sa>y, 

bed 


PROPORTION.  95. 

Hence  a  =  dz^,  h  =  dz^^  c  =  dz,  and  their  sum  is 
d{z^  +  z'^-Yz+  1)=65. 

.*.  d{z'^  +  1)(^  +  1)=  65,  and  since  we  can  make  z  what  we 
please,  the  problem  is  indefinite,  i.e.  it  admits  of  any  number  of 
solutions. 

If  ^  =  2,  the  numbers  are  ^f^,  -\2_^  j2_6^  ^nd  -i/-. 

If  5J  =  I,  the  numbers  are  8,  12,  18,  and  27. 

EXERCISE  VI.  a. 

1.  For  what  value  of  x  will  the  ratio  5  +  x :  8  +  a:  become  5  :  8, 
6:8,  7:8,  8:8,  9:8? 

2.  In  a  city  A  a  man  assessed  for  $  10,000  pays  $72  tax,  and  in 
a  city  B  a  man  assessed  for  1 720  pays  $  4.50  tax.  Compare  the 
rate  of  taxation  in  A  to  that  in  B. 

3.  The  ratio  a  —  l-.h  —  \  is  a,  and  that  of  a  +  1 :  6  +  1  is  /3. 
Find  the  ratio  a  :  6  in  terms  of  a  and  )3. 

4.  Two  men  can  do  in  4  days  what  3  boys  can  do  in  5  days. 
Compare  a  man's  working  ability  with  that  of  a  boy. 

5.  Find  the  number  for  which  the  cube  root  of  its  square  is  to 
the  square  root  of  its  cube  as  m  to  n. 

6.  Given  3^2  +  10a;?/ +  3?/2:  3^2  ^  gx?/ -  3x2  =  2x:  ?/,  to  find 
the  ratio  y  :  x. 

7.  It  a:h  =  h'.c  =  cid^  show  that  a^  -.y^  =  a:c,  and  a^ .  53  _ 
a:  d. 

8.  If  ^{p  +  q):^{p  —  q)=  mm,  find  p  :  g,  and  also  the  dupli- 
cate ratio  of  p  —  q  :  q. 

9.  If  l^-m'^  +  n'^  =  0  and  z  +  m  -  n  =  0,  find  I :  m. 

10.  Find  the  ratio  of  (x  +  h)-'^  —  x'^ :  h  when  h  approaches 
zero. 


,96      GENERALIZED   PROPORTION,    OR   VARIATION. 


GENERALIZED    PROPORTION,    OR    VARIATION. 

85.  Let  x  be  a  variable,  and  let  y  be  connected  with  x 
by  a  constant  multiplier  m,  so  that  y  =  mx.  When  x 
changes  its  value,  becoming  Xi,  say,  y  also  changes  value, 
becoming  2/1,  so  that  y^  =  mx^. 

Dividing  one  equation  by  the  other, 

y  _  mx  __  x^ 
yi     mxi     osi 

Whence  2/  •  !/i  =  ^  •  ^1^ 

i.e.  any  two  values  of  x  and  the  corresponding  values 
of  y  are  in  proportion. 

Also,  if  X  takes  a  series  of  values,  Xi,  x^,  x^,  etc.,  and 
the  corresponding  values  of  y  be  y^,  2/2?  2/3?  ^^^v 

^1*2/1  =  •'^2  •  ^2  =  ^3  •  2/3  =  ^^^* 

The  foregoing  relations  are  indicated  by  saying  that 
y  varies  as  x,  or  x  varies  as  y,  since  the  relation  is 
mutual,  and  they  are  symbolically  expressed  by  writing 
2/  oc  i»,  or  xccy. 

Hence  to  say  that  y  varies  as  x,  is  to  say  that  one  is  a 
constant  multiple  of  the  other,  or  that  any  two  values 
of  X  and  the  corresponding  values  of  y  are  in  proportion. 

86.  li  y  =  n  '  -,  y  varies  as  the  inverse  or  reciprocal 
of  2; ;  or  2/  varies  inversely  as  z. 

If  y  =  n  '  -,  y  varies  directly  as  x  and  inversely  as  z. 

z 

If  2/  =  nxz,  y  varies  conjointly  as  x  and  z. 

Ex.  1.  If  xtxyz,  and  y  varies  inversely  as  2;2^  and  if  2?  =  2  when 
x  =  10,  it  is  required  to  express  x  in  terms  of  z. 


GENERALIZED   PEOPORTION,    OR    VARIATION.      97 


We  have 

x  =  myz,  and  2/cc— • 

z 

And 

10  =  ^,  or  w  =  20. 

2 

...x  =  ?2. 
z 

Ex.  2.  The  velocity  of  a  body  falling  from  rest  varies  as  the 
square  root  of  the  space  passed  over,  and  when  the  body  has  fallen 
16  feet  its  velocity  is  32  feet.  Find  the  relation  between  space  and 
velocity. 

X)  =  m^s,  where  v  =  velocity  and  s  =  space. 

.'.  I?  =  32  and  s  =  16  gives  32  =  4  m. 
,'.  m  =  8,  and  v  =8  Vs,  or  v"^  =  64:S; 
which  shows  that  the  velocity  varies  as  the  square  root  of  the  space 
fallen  through. 

Ex.  3.  The  radius  of  the  earth  is  r,  and  the  attraction  upon  a  body 
without  varies  inversely  as  the  square  of  the  body's  distance  from 
the  centre.  The  number  of  beats  made  per  day  varies  as  the 
square  root  of  the  earth's  attraction  upon  the  pendulum.  How 
much  will  a  clock,  with  a  second's  pendulum,  lose  daily  if  taken  to 
a  distance  r^  from  the  earth's  centre,  r^  being  greater  than  r. 

Let  n  =  the  number  of  seconds  in  a  day  =  86400,  and  let  g  =  the 
earth's  attraction  at  the  surface. 

Then  g  oc— ,  and  n  oc  -.yg. 

.*.  WX-,  and  we  write  n  =  —,  m  being  constant. 
r  r 

Also,  if  n^  be  the  number  of  beats  per  day  made  by  the  pendu- 
lum in  its  new  position, 

Til  =  —  =  n  •  — ,  by  substituting  for  m. 

And  the  clock  loses  n  —  n^  seconds  daily, 

z=z  nil  —-]=  n-  ^^  ~  ^  seconds. 
\        rj  ri 

If  r  =  3960  and  r^  =  3961,  the  loss  is  21.81  sec. 


98      GENERALIZED   PROPORTION,    OR   VARIATION. 


EXERCISE  VI.  b. 

1.  The  space  passed  over  by  a  body  falling  from  rest  varies 
as  the  square  of  the  time,  and  a  body  is  found  to  fall  196  feet  in 
3J  seconds.     Pind  the  relation  between  the  space  and  the  time. 

2.  li  X(xy  and  y  =  3|  when  x  =  6 J,  find  the  value  of  y  when 

3.  y  varies  inversely  as  x^,  and  z  varies  directly  as  x^.  When 
x  =  2f  y  -{-  z  =  340  ;  and  when  x  =  1,  y  —  z  =  1275.  For  what 
value  of  aj  is  ?/  equal  to  z? 

4.  zccu  —  V,  uccx,  and  v  gc  x^.  When  x  =  2,  ^  =  48  ;  and  when 
X  =  5,  ^  =  30.    For  what  value  of  x  is  2;  =  0  ? 

5.  If  x?/  GC  x^  +  y%  and  x  =  3  when  y  =  4,  find  the  relation  con- 
necting X  and  y. 

6.  The  area  of  a  rectangle  is  the  product  of  two  adjacent  sides ; 
if  the  area  is  24  when  the  sum  of  the  sides  is  10,  find  the  sides  of 
the  rectangle. 

7.  If  X  +  2/  oc  X  —  2/,  then  x"^  +  y^cc  xy. 

8.  If  X  cc  y,  show  that  x^  +  y^oc  xy. 

9.  A  watch  loses  2 J  minutes  per  day.  It  is  set  right  on  March 
15th  at  1  P.M. ;  what  is  the  correct  time  when  the  watch  shows 
9  A.M.  on  April  20th  ? 

10.  The  volume  of  a  gas  varies  directly  as  its  absolute  tempera- 
ture, and  inversely  as  its  tension.  1000  cc.  of  gas  at  240°  and 
tension  800  mm.  has  its  temperature  raised  to  300°  and  its  tension 
lowered  to  600  mm.     What  volume  has  the  gas  then  ? 

11.  The  attraction  at  the  surface  of  a  planet  varies  directly  as 
the  planet's  mass  and  inversely  as  the  square  of  its  radius.  The 
earth's  radius  being  3960  miles,  and  the  moon's  1120,  and  the  mass 
of  the  earth  being  75  times  that  of  the  moon,  compare  the  attrac- 
tions at  their  surfaces. 


GENERALIZED   PROPORTION,   OR   VARIATION.      99 

12.  The  length  of  a  pendulum  varies  inversely  as  the  square  of 
the  number  of  beats  it  makes  per  minute,  and  a  pendulum  39.2  in. 
long  beats  seconds.  When  a  seconds  pendulum  loses  30  sec.  per 
day,  how  much  too  long  is  the  pendulum  ? 

13.  When  one  body  revolves  about  another  by  the  law  of  gravi- 
tation, the  square  of  the  time  varies  as  the  cube  of  the  distance. 
The  moon  is  240,000  miles  from  the  earth,  and  makes  her  circuit 
in  27  days.  In  what  time  would  she  complete  her  circuit  if  she 
were  10,000  miles  distant  ? 


CHAPTER  VII. 

Indices  akd  Surds. 

87.  The  index  law  is  the  result  of  the  convention  that 
when p  is  Si  positive  integer,  a^  a  •  a-**  to  p  factors  shall 
be  denoted  by  a^.  And  by  this  law  a^  -  a^  =  a^+*,  p  and  q 
both  being  positive  integers. 

Now,  if  algebra  is  to  be  consistent  with  itself  we  can 
have  only  one  index  law,  whatever  p  may  denote,  and 
instead  of  making  a  new  convention  we  must  interpret 
in  conformity  with  this  index  law  the  cases  in  which  p 
is  not  positive  and  integral. 

The  interpretation  of  p  zero,  or  negative  and  integral, 
is  given  in  Art.  38.     We  deal  here  with  p  fractional. 

(1)  If  p  =  g  =  i,  a^  .  a'^  =  a*  .  a*  =  a^  =  a. 

Therefore,  a^  is  the  same  as  ^a,  the  meaning  of  which 
is  fully  given  in  Arts.  47  and  48. 
Similarly,  a^  .  a^  .  a^  =  a^"^  3  +  3  _-  ^^ 

And  a^  means  that  a  is  to  be  separated  into  three 
identically  equal  factors,  and  that  one  of  these  is  to  be 
taken. 

By  an  obvious  extension, 

-        -        -  .  n       .  — I 1 !-•••  ton  terms 

a".a"-a"--- to  n  factors  =  a"   "   "  =  a; 

1 
and  a**  tells  us  to  separate  a  into  n  identically  equal 
factors,  and  take  one  of  these  factors. 
100 


INDICES.  101 

This  factor  is  called  the  nth  root  of  a,  and  is  often 
written  ^a,  the  letter  or  figure  being  written  in  the 
symbol  -y/  in  all  cases  where  n  is  not  2.  Thus  -^/a  de- 
notes the  cube  root  of  a,  etc. 

Ill  1^ 

(2)  a""  -a""  -a""  "•  to  m  factors  becomes  (a")"*  when  we 
1 
consider  (a")  as  a  single  quantitative  symbol;   and  it 

becomes  a^  by  the  index  law. 

1  m 

Therefore,  (a")"*  =  a";  and  either  of  these  expressions 
denotes  that  the  nth  root  of  a  is  to  be  raised  to  the  mth 
power;  or  that  the  mth  power  of  a  is  to  have  its  nth 
root  taken. 

Thus  (64)*  =  (64^)2,  or  (642)^  =  16. 

Also,  writing  p  for  -,  and  -  for  m,  gives  — 
n  q 

(a^)«=:a^;  or  (a^) » =  a*  =  (a*)^. 


(3)  Making  p=:-  in  (2)  gives  — 


11        J.  11 

or  V^=  V^  =  V^. 

Thus  to  get  the  6th  root  of  a  quantity  we  may  first 
take  the  square  root  and  then  the  cube  root  of  the  result, 
or  we  may  first  take  the  cube  root  and  then  the  square 
root  of  the  result. 


Ex.  1.  To  simplify  f^V^".  C^Y^'J^Y 


102  INDICES. 

This  becomes    (xm-n^m+n  .  (^^n-iy+l  .  (^y.l~my+m 

—  x"*^-"^ .  x"'^-*^  •  x^^-""^  =  x'"^-«^+»^-^^+^^-"»^  =  x'^  =  1. 
Ex.2.    To  simplify        ^' ■  ^'^  ■  ^(i"-\ 
This  becomes 

23  .  26-3n  .  24n-4  .  3-2+n  .  3-2n  .  3-3+3n  —  25+n  .  3-5  _  ?!!lL. 

EXERCISE  VII.  a. 

-      1——      3  3 

1.  Simplify    (a«.a     "•  a2)-^(«2 .  ^j.  ^-i)^ 

2.  Simplify    V^*  •  ^^^ .  ^2 .  ^-Ki+n). 

3.  Simplify    (ic2)2.  (xJ)n-i  ^(x«)i  .  (x2«)«. 

4.  Simplify    a-«+i  •  a"^"  •  a^ -^  «-«.  a2.  «l 
6.    Simplify    (x^+«  -^  5C«)^  ^  (ic«  -^  x^-py-^. 

p   q 

6.  Simplify    V{(«~^"*  "^  2>-2«)mj«. 

7.  Write  the  square  of  av^2  _  (j-N/2, 

8.  Multiply  a3  _  q^I^t  4.  ^2  i3y  ^ it  +  5?. 

9.  Express  the  relation  x  =  ^<  ~  ~  "^  A/ (t  ~  r~  )  I '  ^^  ^^  to 
be  free  from  irrationals. 


10.   Express  the  relation 


Va  +  X  +  Va  —  aj 


,     .      .        ...  Va2-x2 

to  be  free  from  irrationals. 

11.   Find  X  when  a* .  a="  -4-  a*~^  —  ^^j. 


12.    Find  n  when  2" .  2«-i  •  2«+i  =  2i-"  •  2i+«  •  2i-». 


=V(:-^3- 


SURDS.  103 

n 

13.  Find  n  when  Qs  .  3»»-i  =  6« .  2-^  •  3«+i. 

14.  Find  the  relation  between  m  and  n  when  a"*"  =  (a™)**. 

n 

15.  Find  n  when  4"  •  22«-i .  8i-»  =  42  •  2-»»  •  163. 

16.  Divide  x^  -  Sx'^  +  2  -  x-^  +  x-^  by  ic  -  3  -  2 a;-i,  giving 
quotient  and  remainder. 

17.  If  fx  =  i(a'  +  a-*),  and  0x  =  |(a*  -  a-*),  show  that  — 
i-    (fxy^-(<pxy  =  l.  iii.   2(0aj)2=/2x-l. 

ii.   2(/x)2  =  1  +  /(2 X),  iv. .  0(2  X)  =  2(0x) .  (/x). 

18.  If /x  =  J(a^  + a~**)  and  0x=i(a»*  -  a-**),  show  that  — 

i-     (/x)2+(</>X)2  =  l.  ii.     0(_x)  =  -(0X). 

iii.  /x=/(-x). 

SURDS. 

88.  A  surd  is  the  incommensurable  root  of  a  com- 
mensurable number  (Chrystal). 

Thus  ^2j  -^5j  3*,  etc.,  are  symbolic  expressions  for 
surds.  The  arithmetical  extraction  of  the  roots  indicated 
would  approximate  to  the  numerical  values  of  the  surds. 

The  expression  ^4  is  an  integer  under  a  surd  form. 

Many  known  incommensurables  are  not  surds,  and 
some  of  them  are  not,  as  far  as  we  know,  due  to  any 
finite  combination  of  surds.  As  examples,  we  have 
3.1415926  •••,  which  is  the  ratio  of  the  circumference  of 
a  circle  to  its  diameter,  and  which  is  usually  denoted  by 
the  greek  letter  tt;  and  2.7182818  •••,  which  is  the  base 
of  the  natural  logarithms,  and  which  is  usually  denote^ 
by  c  or  e. 

An  expression  such  as  V2  +  ^2  is  a  surd  expression, 
but  is  not  a  surd  according  to  definition. 


104  SURDS. 

89.  Let  z  and  u  be  any  real  positive  quantities,  n  being 
an  integer. 

Then  a;**  will  pass  through  all  values  from  0  to  +  oo, 
if  X  passes  through  all  values  from  0  to  +  oo.  There- 
fore for  some  positive  value  of  x,  a;"  =  z,  and  x  =:  -^z. 

That  is,  every  quantity  between  0  and  +  oo  has  a 
positive  real  nth  root.  This  is  the  arithmetical  root, 
and  is  the  one  with  which  we  are  principally  concerned.    , 

If  n  is  even,  n  =  2?/i ;  and  2;"  =  2;^"*  =  («'")*  And  since 
a  square  root  has  two  signs  — 

.-.  every  even  root  of  any  quantity  has  two  values 
differing  only  in  sign. 

90.  Let  X  be  negative  and  n  be  an  odd  positive  integer. 
Then  a;*"  is  negative,  and  passes  from  0  to  —  00,  while  x 
passes  from  0  to    —  00. 

Therefore,  for  some  value  of  a?,  »"  =  —  z,  and  x=-\/ —z. 
That  is,  every  real  negative  quantity  has  a  real  nega- 
tive nth  root,  when  n  is  odd. 

Thus,  every  positive  real  quantity  has  two  real  square 
roots,  fourth  roots,  sixth  roots,  etc.,  and  one  real  cube 
root,  fifth  root,  etc.  And  every  real  negative  quantity 
has  one  real  cube  root,  fifth  root,  etc.,  and  no  real  square 
root,  fourth  root,  etc. 

91.  Let  a  be  any  real  positive  quantity,  and  let  r  be 
its  arithmetical  cube  root. 

Then,  since  w'  =  w^  =  1,  r^  =  wV  =  wV  =  a.  And  tak- 
ing the  cube  root,  r,  cor,  and  coV  are  each  cube  roots  of  a. 
And  the  three  cube  roots  of  a  are  -y/a,  (u-^/a,  in^^a, 
where  -^a  is  the  arithmetical  cube  root. 

Thus  the  three  cube  roots  of  27  are  3,  3  <u,  3  w^.     Hence 


SURDS.  105 

every  real  quantity  has  one  arithmetical  cube  root,  and 
two  complex  cube  roots. 

92.  Sards  which  are  reducible  to  the  same  surd  factor 
are  similar  ;  otherwise  they  are  dissimilar. 

To  reduce  a  surd  to  its  surd  factor  we  proceed  as 
follows : 

Decompose  the  number  into  its  prime  factors,  and 
then  (1)  for  a  quadratic  surd  take  out  the  largest  square 
factor  possible.     The  remaining  factor  is  the  surd  factor. 

Ex.  1.  V(1350)  =  V(2  .  33 .  52)  =  3  .  5V(2  •  3)  =  15  V6, 
and  6  is  the  surd  factor. 

(2)  For  a  cubic  surd  take  out  the  largest  cube  factor 
possible.     The  remaining  factor  is  the  surd  factor. 

Ex.  2.  -^(9720)  =  ^(2^.  3^ .  5)  =  2  •  3^(3^ .  5) 

=  6^(45),  and  45  is  the  surd  factor. 

93.  Surds  of  the  same  order  are  added  by  reducing 
them  to  their  surd  factors,  and  adding  the  coefficients  of 
similar  surds. 

Ex.  3  V2  +  V^^  +  2  V12  -  V^S 

=  3V2  +  3V2  +  4V3   -4V3  =  6V2. 

With  dissimilar  surds,  or  with  surds  of  different  orders, 
the  addition  and  subtraction  can  only  be  indicated  by 
connecting  them  with  the  proper  signs. 

94.  When  a  fraction  contains  a  surd  expression  as 
denominator,  it  becomes  necessary,  for  the  sake  of  ease 
in  calculation,  to  so  transform  the  fraction  as  to  make 
its  denominator  rational.     This  is  effected  by  multiply- 


106  SURDS. 

ing  both  parts  of  the  fraction  by  some  expression  which 
will  make  the  denominator  rational.  Such  a  multiplier 
is  called  a  rationalizing  factor  of  the  denominator. 

1          1/2/2 
Ex.  1.   — = ^^^=^^^,    and  the   denominator  is  rational, 

and  -^2  is  the  rationalizing  factor. 

Ex.2.  2  ^2^2^.  V^^2J/^J^^/4^  andthedenom- 
inator  is  rational,  and  ■^2'^^S  is  the  rationalizing  factor. 

95.  Since  {^a -{- ^b){-y/a— ^b)  =  a  —  b,  which  is 
rational,  the  expression  y'a  —  ^b  is  a  rationalizing 
factor  for  -y/a  +  -^/b ;  and  reciprocally  y'a  +  -^b  is  a 
rationalizing  factor  for  -^a  —  -y/6. 


Ex.  1.    To  rationalize  the  denominator  of 


3V2 
^ys  -  v/2' 


Multiplying  by  ^^  +  V^  gives  ^V^  +  ^  or  3(^6  -  2). 
V^  +  -^2  1 

14-2/2 
Ex.  2.   To  rationalize  the  denominator  of  -^ — ^ — 

H-V2 

Multiplying  by  Liza^  gives  ~^~^^,  or  3  -  V2. 
1  —  •y/2  —  1 

In  the  present  case  it  is  better  to  change  the  order  of  the  denom- 
inator, writing  it  -^2  +  1,  since  that  gives  a  +  quantity  in  the 
denominator  of  the  result,  and  does  not  affect  the  value  of  the 
fraction. 

Ex.  3.  To  rationalize  the  denominator  in 


1  +  V2  +  V^ 
This  may  be  effected  at  two  steps. 

Multiply    by    ^  +  V^  -  V^     and    we    get     ^  +  V^  -  V^     or 

1  +  V^  -  V3.    And  this  multiplied  by  ^  gives  V^  +  ^'^g. 
2-\/2  •y/2  4 


SURDS.  107 

Ex.  4.  To  rationalize  the  denominator  in  — y  ^  4- ^/g 

V2  -  V3  +  V^ 

Multiplying  by  V^  -  ^3  -  V^  gj^^s  ~^~V^^~V^^  And 
multiplying  by  :^  gives  "  V^ -j/^Q  -  V^Q^ 

or  ye  +  2  V15  +  3  yio 

12 

A  trinomial  quadratic  deDominator  may  also  be 
rationalized  by  a  single  multiplication,  as  we  proceed  to 
show. 

96.  To  find  a  rationalizing  factor  for  ^a  -\-  -yjh  +  yc, 
where  the  surds  are  all  dissimilar. 

Let  Va  =p,  V^  =  q,  ^c  =  r. 

Then     {p -\.q^r){p  ^  q  —  r)  =p^  +  g^  _  r^  +  2pg ; 

and  (jp^  +  g^  —  r^  +  2pq)  (p^  +  g^  —  r^  —  2j9g) 

=  (i>^  +  g^  —  7-^)  ^  —  4p  V- 

In  this  final  product  p,  g,  and  r  appear  only  in  even 
powers,  and  the  product  is  accordingly  rational. 

The  rationalizing  factor  is  the  co-factor  of  (p  +  g  +  ^); 
that  is,  (i>  +  g  —  ^)  (i>^  +  g^  —  r^  —  2pg) . 

This  factor  reduces  to  Sp^  —  '^p\  +  2pqr\  or  restoring 
-^/a,  y'ft,  and  y'c,  to 

2 (a  —  &  —  c)  Va  +  2  V(^^0  5 
and  this  is  the  rationalizing  factor. 

The   rationalized   expression  is    Sa^  —  2  Sa^,  and   the 

fraction  becomes 

2(a  —  &  —  c)  yg  +  2  V(a&c) 
2a2-2  2a& 


108  SURDS. 

1 


Ex.  To  rationalize  the  denominator  of 


1  +  V2  +  V3 
a  =  1,  6  =  2,  c  =  3. 

S(a  -  &  -  c)  V«  +  2^(abc)  =  -2V2  -  4  +  2^6, 
and  2a2  _  2  2a6  =  -  8. 

/.  the  fraction  becomes   ^     +        V  .. 

4 

In  using  the  form  :S(a  —  &  —  c)  -y^a  -\-  2y'(a5c),  if  any 
of  the  terms  are  negative,  the  proper  signs  must  be 
attached  to  the  parts  involving  the  roots  only. 

Thus  for  -yy2  —  -y/S  + 1  the  rationalizing  factor  is 
^  2  V2  -  4  -  2  V6. 

For  V7  —  V^  —  V^  *^®  factor  is 

2  V7  +  6  V3  +  8  V2  +  2  Vi2. 

The  rationalized  result  2a^  —  2  2a&,  not  involving  any 
roots,  is  not  affected  by  the  signs  of  the  terms,  and 
remains  the  same  for  all  signs. 

97.   The  expression   S(a  —  6  —  c)  ^a  +  2-\/abc  factors 

into  (  v«  —  V^  +  V^)  (  v^  +  v^  —  V^)  ( v^  —  v^  —  V^)  • 

This  result,  although  not  convenient  in  practice,  is 
interesting  as  showing  the  constitution  of  a  rationalizing 
factor. 

For  the  three  factors  are  each  derived  from  the  expres- 
sion to  be  rationalized,  -^/a  -f-  -y/b  +  -^/c,  by  keeping  one 
term  unchanged  and  giving  to  the  other  two  all  possible 
variations  of  sign.  So  that  of  the  four  trinomial  expres- 
sions v^  +  V^  +  V^j  v^  +  v^  — v^?  v^  — v^+v^^ 

and  -^a  —  -y/b  —  ^c,  the  product  of  any  three  is  a  rational- 
izing  factor  of  the  fourth. 

This  principle  admits  of  great  extension. 


SURDS.  109 

98.    To  find  a  rationalizing  factor  for  -^a  -f-  -^b. 

Let  a  =  p%  and  6  =  ^. 

Then       (p  +  q)  (p^  —pq  -{- q^)  =  p^  +  q^  =  a -}- b. 

.*.  p^—pq-\-  q^,  or  -^a^  —  -y/ab  +-^b^  is  a  rationalizing 
factor  for  -^a  +  ^&. 

Similarly,  -^a^  +  ^a6  +  -^/ft^  is  a  rationalizing  factor 
for  ^a  -  -^6. 

Ex.  1.  To  rationalize  ^a  +  ^6. 

( V«  +  v^^)( V«  -  ^&)  =  a  -  ^&2=  ^a^  -  ^&2 ; 
and  this  being  the  difference  between  two  cube  roots,  the  rational- 
izing factor  for  it  is  — 

or  a2  +  ab^  +  &^, 

and  the  factor  required  is  — 

(  V«  -  W(«^  +  <^Vb^  +  b^b). 
The  rationalized  expression  is  a^  —  b^. 

EXERCISE  VII.  b. 
1.    Simplify  — 

i.   (3+V5)(2-VS).  V.  5 V2(3 V4  +  6 V2). 

ii.    U+:^\(-3  +  y^\'         vi.   \/48^+6\/75a. 
iii.   (4Vi+ViXVH2Vi).        ""•  V8-7V18+5V72-V50. 


iv.    •v/{v^8¥3}.  viii.   ^/iSa^b'^ -\-  V^Oa^b^. 

2.    Rationalize  the  denominators  of — 

i.  ^_.  ii.  :v^3^.  iii.   ^V^-^. 

V8^  2  -  V3  3  -  2V2 


110  SURDS. 

iv     V2(V5-1).  ^.     5  4-2V2-3V.3-V6. 

2V5TV5  1+V2-V3 


V. 


3  +  2V2-3V3-2v/6 


1  +  V2  -  V^  -  V^  1  -  V2  -  V3 

8.  Which  is  the  greater,  ^^/^jzi  or  ^V^~^  ? 

4.  Which  is  the  greater,  ^^<!- — '^ —  or  -^'  ~  ,  and  what  is  their 
difference?  >/^  >/^ 

5.  Rationalize  the  denominator  of  — '^ 

^2+^3 

6.  Rationalize  the  denominator  of  — "-^ 

V2+^2 

99.  The  following  theorems  are  important  in  relation 
to  quadratic  surds. 

(1)  The  product  of  two  dissimilar  surds  cannot  be 
rational. 

For,  if  p  and  q  be  their  surd  factors,  p  and  q  do  not 
contain  the  same  factors,  and  hence  pq  is  not  made  up 
of  square  factors  only. 

Therefore  Vpq  is  not  rational. 

(2)  A  quadratic  surd  is  not  the  sum  or  difference  of 
a  rational  quantity  and  a  surd. 

For,  if  ^a  =  b  ±  -y/c,  a  —  b^  —  c  =  ±2b  ^c,  by  squaring. 
But  a  —  b^  —  c  is  rational,  and  therefore  y^c  is  rational, 
which  is  contrary  to  hypothesis. 

Cor.  The  sum  or  difference  of  two  quadratic  surds  is 
not  rational. 

(3)  A  surd  is  not  the  sum  or  difference  of  two  dis- 
similar surds. 


SURDS.  Ill 

For,  if  ■^^a  =  ^h  +  -y/c,  a  —  b  —  c  =  2^ he,  by  squaring. 
But  by   (1),  V^c  is  not   rational,  while   a  —  b  —  c  is 
rational.     .*.  etc. 

(4)  li  x-{-  ^y  =  a  +  ^h,  where  x  and  a  are  rational, 
then  x  =  a  and  y  =  b. 

For,  X  —  a  =  ^/b  —  -y/y ;  and  if  a?  —  a  has  any  value, 
it  is  rational,  and  is  equal  to  the  difference  of  two  surds, 
which  is  impossible  by  (2),  Cor. 

.'.  x  —  a  =  0,  or  x=a  and  b  =  y. 

It  may  be  well  to  remark  here  that  with  an  equation, 
such  as  x  =  a+  -y/b,  we  do  not  mean  to  assert  that  x  has 
not  an  exact  finite  value,  but  merely  that  its  value  is  not 
that  of  a  quadratic  surd ;  or,  in  other  words,  the  square 
of  x  is  not  rational.  So  also,  if  a;  —  a  =  ^b—-yj'yj  where 
6  and  2/  are  not  equal,  x  —  a  has  a  real  finite  value,  but 
this  value  is  not  rational,  and  its  square  is  not  rational. 
i.e.  x  —  a  is  not  a  surd.  Hence,  if  x  and  a  be  both 
rational,  x^a  cannot  have  any  value,  since  such  a  value 
must  of  necessity  be  rational. 

100.  The  results  of  Art.  99  enable  us  to  separate  a 
surd  expression,  such  as  a  +  y 6  or  ^a  +  ^b,  into  two 
identical  factors,  i.e.  to  find  its  square  root. 

Let  one  factor  be  -y/x  +  -y/y. 

Then     a  +  -sJb  =  {-y/x-\-^yy=x-\-y -\-2^xy, 

.*.  Art.  99  (4),     aj  +  2/  =  a;  and  Axy  =  b. 

But        x--y  =  ^\(x-\-yy-4xy\=^\a'-bl 


x  =  ilx-}-y-}-x-yl  =  i\a-h^{a'-b)l 


y  =  i\x  +  y-x-y]  =  i\a-^{a'-b)\ 


112  SURDS. 

.-.  V^  +  V^  =  V H«  +  Vo^"^  +  VH«  -  Va2  -  6) , 
which  is  the  required  factor. 

This  factor,  upon  being  squared,  will  reproduce  the 
expression  a  +  -yjl) ;  but  the  case  of  practical  utility  is 
that  in  which  a^  — 5  is  a  complete  square.  Denote  it 
byc^. 


Then  V^  +  V2/  =  Vi(a  +  c)-^^^{a  -  c). 

Ex.  1.  To  find  the  square  root  of  3  +  2^2. 
Here  a  =  3  and  6  =  8,  and  a^  —  6  =  c^  =  1,  a  complete  square, 
whose  root  is  1. 

/.  Va^  +  V2/=VF4+VF2=V2  +  1. 
Ex.  2.   To  find  V{^^  ■\-n-2  n^n}. 
Here  a  =  n^  +  w,  6  =  4  n^,  c^ri^  —  n, 

}(a  +  c)=n2,  i(a  — c)=w. 

.'.  V{^^  -\-  n  —  2  ny/n}=  n  —  ^n. 

The  sign  —  before  -^n  in  the  result  is  indicated  by  the  same 
sign  before  2  Uy/n  in  the  original. 

101.  An  expression  of  the  form  a  +  y^^ -f  y'c  +  V^ 
can  have  its  root  found  when  the  expression  is  a  com- 
plete square. 

Assume  V{^  + V^  +  V^  + V^l  =  V^  +  V2/  + V^- 
Squaring, 

.-.  a=:x-\-y -\-z,  b  =  4:yz,  c==4:zx,  d  =  4:xy. 


SURDS.  113 


"^-^"'■"HirMrHir^. 


Ex.    Root  of  35  +  4^15  -  6^10  -  12  V^- 
Here  b  =  240,  c  =  360,  d  =  864. 

.    *  1/360  X  864\      4 1/864  x  240\      ^//240  x  360\ 
**   \V  4x240'j      \V  4x360  J      \\  4  x  864  j 

=  V18  ±  V^2  ±  V5 
=2V3±3V2±V5. 

The  signs  are  then  to  be  determined.  It  is  readily  seen  that 
they  must  be 

2V3-3V2+V'5 
in  order  to  give  those  of  the  original. 

Moreover,         (2  V3)2  +  (3  V2)2  +  (^6y  =  35. 

The  original  expression  should  be  reproduced  by  squar- 
ing this,  and  that  is  the  only  absolute  test  that  we  have 
the  true  root. 

EXERCISE  VII.  C. 

Find  the  square  roots  of  the  following  perfect  squares  from  1  to 
11  inclusive. 


1.    8  +  2^15.      2.    3  +  41     3.   4+2^3.     4.    l  +  2uVl-u^. 


5.    -2i.  6.    2^2  +  2(x  -  2/)\/x2  -  y^  -2xy. 

7.  9-4V2  +  4V3-2V6.  ^q    4^10  +  8 

8.  25  +  10  V6.  ^^^^  ~  ^ 


9.    I  +  V2.  11.  a2  _  2  +  a\/^^2Tr4. 

12.    Simplify      1 ^— +  ^ 


V'(16  +  2  V63)       V(16  -  2  V^'^) 

13.  Simplify      ^(3  +  VO  -  p^)  +  V(3  -  Vo  -  p^). 

14.  If  a'^d  =  be,  then  V(^  +  V^  +  V^  +  V^)  ^^^  he  put  in  the 
form  (V«5  +  V2/)(V^+\/^)- 


CHAPTER  VIII. 

Concrete   Quantity.  —  Geometrical   Interpre- 
tations.—  The  Graph. 

102.  A  concrete  number  depends  upon  a  concrete 
unit  which  gives  to  the  number  its  name  and  character. 
Thus  8  dollars  is  8  of  the  concrete  units  called  a  dollar. 
So  a  hours  is  a  times  the  concrete  unit  known  as  an 
hour. 

The  symbolism  of  algebra  applies  to  these  abstract 
numbers,  as  coefficients  of  concrete  units,  but  the  inter- 
pretation is  affected  by  the  nature  of  the  concrete  unit. 

Operations  with  concrete  quantities  are  in  general 
subject  to  the  two  following  laws : 

(1)  To  multiply  2  hours  by  3,  or  3  hours  by  2,  gives 
6  hours;  but  to  multiply  2  hours  by  3  hours  has  no 
meaning. 

Hence,  concrete  units  have  no  product,  i.e.  they  do 
not  admit  of  being  multiplied  together. 

Hence,  also,  concrete  units  have  no  powers  and  no 
roots. 

(2)  Again,  3  hours  and  4  hours  make  7  hours,  and 
5  minutes  and  8  minutes  make  13  minutes,  but  3  hours 
and  4  minutes  do  not  make  7  hours  or  7  minutes. 

Hence,  concrete  quantities  are  added  only  when  of  the 
same  name,  and  then,  by  adding  the  coefficients  of  the 
concrete  unit. 

114 


CONCRETE   QUANTITY.  115 

An  important  apparent  exception  to  the  foregoing  laws 
occurs  in  units  of  length ;  this  will  be  fully  dealt  with 
in  connection  with  geometrical  interpretations. 

Any  other  apparent  exceptions  are  easily  explained. 

103.  In  many  cases  of  the  interpretation  of  concrete 
results  negative  quantity  has  a  special  significance. 

If  an  idea  which  can  be  denoted  by  a  quantitative 
symbol  has  an  opposite  so  related  to  it  that  one  of  these 
ideas  tends  to  destroy  the  other  or  to  render  its  effects 
nugatory,  these  two  ideas  can  be  algebraically  and 
properly  represented  only  by  the  opposite  signs  of 
algebra. 

If  a  man  buys  an  article  for  h  dollars,  and  sells  it  for 
s  dollars,  his  gain  is  expressed  by  s  —  h  dollars.  So 
long  as  s  >  6,  this  expression  is  +,  and  gives  the  man^s 
gain. 

But  if  s<6,  the  expression  is  — .  It  denotes  that 
whatever  his  gain  is  now,  it  is  something  exactly  opposite 
in  character  to  what  it  was  before.  And  as  he  now  sells 
for  less  than  he  buys  for,  he  loses.  In  other  words,  a 
negative  gain  means  loss. 

Thus  gain  and  loss  are  ideas  which  have  that  kind  of 
oppositeness  which  is  expressed  by  oppositeness  in  sign. 
If  a  man  gains  +  a  dollars,  he  is  so  much  the  wealthier ; 
if  he  gains   —  a  dollars,  he  is  so  much  the  poorer. 

Whether  gain  or  loss  is  to  be  considered  positive 
must  be  a  matter  of  convenience,  but  only  opposite  signs 
can  denote  the  opposite  ideas. 

104.  Among  the  ideas  which  possess  this  oppositeness 
of  character  are  the  following ; 


116  CONCRETE  QUANTITY. 

(1)  To  receive  and  to  give  out;  and  hence,  to  buy  and 
to  sell,  to  gain  and  to  lose,  to  save  and  to  spend,  etc. 

(2)  To  move  in  any  direction  and  in  the  opposite  direc- 
tion; and  hence,  measures  or  distances  in  any  direction 
and  in  the  opposite  direction,  as  east  and  west,  north 
and  south,  up  and  down,  above  and  below,  before  and 
behind,  etc. 

(3)  Ideas  involving  time  past  and  time  to  come ;  as  the 
past  and  the  future,  to  be  older  than  and  to  be  younger 
than,  since  and  before,  etc. 

(4)  To  exceed  and  to  fall  short  of;  as,  to  be  greater 
than  and  to  be  less  than,  etc. 

Ex.  1.  A  man  goes  20  miles  north,  then  15  miles  south,  then  8 
miles  south,  then  12  miles  north,  and  lastly  18  miles  south.  Where 
is  he  with  respect  to  his  starting-point  ? 

Denoting  north  by  + ,  south  becomes  — . 

The  traveller  has  gone  +20  —  15  —  8  +  12  —  18  miles,  or  —  9 
miles. 

That  is,  he  is  9  miles  from  his  starting-point,  and  the  sign  — 
shows  that  he  is  south  of  it. 

Ex.  2.  A  man  invests  $10,000.  On  J  of  his  investment  he 
gains  25  % ;  on  |  he  loses  20  % ;  and  on  the  remaining  portion  he 
gains  2  %.     What  per  cent  does  he  gain  upon  the  whole. 

Let  gain  be  +,  and  let  a  denote  the  amount  invested.  His 
gain  %  is 

fa     25^_2_a    ^  ,  7a      2  \  .     a   _      ^  ^  p. 
U  *  100       5   *  100      20  *  lOOy  '  100  '°  '^' 

Therefore,  his  gain  is  —  l2^o%'  ^^  ^^  loses  l2^o%* 

Ex.  3.  A  goes  a  miles  an  hour  and  B  b  miles  an  hour  along  the 
same  road,  and  A  is  m  miles  in  advance  of  B.  Wnen  and  where 
will  they  be  together  ? 


CONCRETE  QUANTITY.  117 

Let  t  be  the  time  in  hours,  and  j?  be  the  distance  in  miles,  from 
B's  present  position  to  the  point  of  meeting. 

hm 


Then  t  =  _i?L_,  and  p  = 


b  —  a 


1.  Suppose  a,  b,  and  m  to  be  all  positive. 
As  a  and  b  may  have  any  values,  — 

(a)  Let  a  =  b.     Then  t  and  p  are  both  oo. 

Now  when  a  =  b,  A  and  B  are  travelling  at  the  same 
rate  in  the  same  direction,  and  the  values  of  t  and  p  tell 
us  that  they  will  be  together  after  an  infinite  time,  and 
at  an  infinite  distance  from  the  present  position  of  B, 
I.e.  that  they  will  never  be  together.  It  must  be  noticed 
that  this  is  the  only  way  in  which  the  symbols  of  algebra 
can  answer  the  question  proposed  with  these  conditions. 

(b)  Let  a<b.  Then  as  t  and  p  are  both  positive  and 
finite,  the  men  will  be  together  at  some  time  in  the 
future,  and  at  some  distance  in  the  positive  direction 
measured  from  B's  present  position. 

(c)  Let  a  >  &.     Then  t  and  p  are  both  negative. 
This  tells  us  that  A  and  B  will  be  together  at  some  time 

in  the  past,  i.e.  they  have  already  been  together ;  for  as 
t  denotes  time  to  come,  —  t  must  denote  time  past,  and 
their  point  of  meeting  is  at  some  distance  in  the  nega- 
tive direction  from  B's  present  position. 

2.  Let  a  be  negative.  Then  A  is  coming  backwards 
to  meet  B,  and  they  will  meet  at  some  time  in  the  future, 
and  at  some  place  in  the  positive  direction  from  B's 
position. 

Other  variations  of  signs  must  be  left  to  the  explana- 
tion of  the  reader. 


118  CONCRETE  QUANTITY. 

Ex.  4.  A  and  B  both  have  cash  in  hand  and  both  owe  debts. 
B's  cash  is  10  times  his  debt.  If  B  pays  A's  debt,  his  cash  will  be 
2|  that  of  A's,  and  if  A  pays  B's  debt,  his  cash  will  be  ^^  of  B's. 
When  all  debts  are  paid,  both  together  have  $  3400.  How  much 
has  A  after  his  debts  are  paid  ? 

Let  A's  cash  =  a,  and  B's  cash  =  b.  Since  debt  is  opposite  to 
cash  capital,  B's  debt  is capital. 

Then  A's  debt  =  3400  +  ^-a-b  capital. 
10  ^ 

After  paying  B's  debt  A  has  a left,  and  this  is  equal  to 

2%  b,     .-.  a  =  i  b. 

After  paying  A's  debt  B  has  6  -  ^3400  +  ^-a-b\  left,  and 
this  is  equal  to  2|  a.  ' 

:,^Ab  =  b-  3400  -  A  +  5  +  5. 
5     4  10     4 

Whence  b  =  4000  =  B's  capital, 

and  —  400  =  B's  debt,  as  capital. 

.-.  B  is  worth  3600  when  his  debts  are  paid,  and  A  is  worth 
—  200  when  his  debts  are  paid. 

That  is,  A  owes  200  dollars  more  than  he  is  worth. 

105.  The  examples  of  the  preceding  article  illustrate 
the  fact  that  a  literal  algebraic  solution  of  a  concrete 
problem  is  not  a  solution  of  a  particular  problem,  although 
put  in  a  particular  form,  but  of  every  problem  belonging 
to  a  group  of  which  the  particular  one  may  be  taken 
as  a  representative.  This  group  includes  all  problems 
derivable  from  the  particular  one  by  (1)  varying  the 
magnitudes  of  the  numerical  quantities  concerned,  and 
(2)  changing  ideas  which  admit  of  it  into  their  opposites, 
provided  always  that  such  changes  do  not  render  the 
problem  unintelligible. 


CONCRETE  QUANTITY.  119 

When  such  a  literal  result  is  interpreted  in  language 
as  general  as  possible,  it  becomes  a  rule;  but  it  seldom 
happens  that  any  rule  of  arithmetic  or  geometry  can  be 
as  broad  and  representative  as  the  literal  expression  from 
which  it  is  derived,  and  of  which  it  professes  to  be  the 
interpretation. 

EXERCISE  VIII.  a. 

1.  A  man  buys  m  articles  at  b  dollars  per  article  and  sells  them 
at  s  dollars  per  article  ;   what  is  his  gain  per  cent  ? 

Interpret  the  result  (i.)  when  he  receives  the  articles  as  a  gift ; 
(ii.)  when  he  gives  the  goods  away  ;  (iii.)  when  ft  >  s. 

Show  that  a  person  in  dealing  may  gain  any  finite  percentage, 
but  that  he  cannot  lose  more  than  100  per  cent. 

2.  One  carriage  wheel  is  m,  and  the  other  n  feet  in  circumfer- 
ence. How  far  has  the  carriage  gone  when  one  wheel  has  made  r 
revolutions  more  than  the  other. 

Interpret  when  m  =  n. 

3.  A  has  a  dollars  more  than  B,  but  if  B  gives  A  b  dollars,  A 
will  have  twice  as  much  as  B.     How  much  has  each  ? 

Interpret  when  (i.)  a  is  negative;  (ii.)  b  is  negative;  (iii.)  is 
there  any  arithmetical  interpretation  for  both  a  and  b  negative  ? 

4.  A^s  age  exceeds  B's  by  n  years,  and  is  as  much  below  m  as 
B's  is  above  p.     Find  their  ages. 

Interpret  when  n  >  m  +  p. 

5.  The  hands  of  a  clock  go  around  in  the  same  direction  in  a 
and  b  hours  respectively.  If  they  start  from  the  same  point 
together,  when  will  they  be  again  together  ? 

Adapt  your  result  to  the  case  where  the  hands  move  in  opposite 
directions. 

6.  A,  By  Cj  D  are  four  equidistant  fixed  points  in  line ;  find  a 
point  0,  in  the  line,  for  which 

3^O+5SO4-2CO  +  6Z)O  =  0. 
On  which  side  (the  ^  or  i>  side)  of  the  middle  is  the  point  0  ? 


120  GEOMETRICAL  IKTERPRETATIONS. 

7.  Two  circles  have  tlieir  radii  r  and  r^  and  their  centres  lie  on 
a  fixed  horizontal  line.  If  r>ri  and  d  is  the  distance  between 
centres,  how  are  the  circles  relatively  situated  when  — 

i.  d  =  r  -\-  r^f  iii.   d  =  —r  +  r^, 

ii.   d  =  r  —  r^,  iv.  d  =—  r  —  r^? 

8.  A  can  run  a  feet  in  b  seconds,  and  B  can  run  b  feet  in  a  sec- 
onds.   Express  the  ratio  of  A's  speed  to  B's. 

9.  Two  ropes  are  in  length  as  4:5,  and  6  feet  being  cut  from 
each,  the  remainders  are  as  3  :  4.    Find  their  original  lengths. 

10.  The  lengths  of  two  ropes  are  as  a :  6,  and  c  feet  being  cut 
from  each  they  are  as  a^-.b^.    Eind  their  original  lengths. 

Interpret  when  c  is  negative. 

11.  A  river  flows  4  miles  an  hour ;  a  boat  going  down  the  river 
passes  a  certain  point  in  20  seconds,  and  in  going  up  it  takes  30 
seconds.  Eind  the  speed  of  the  boat  in  still  water.  Also  the 
length  of  the  boat  in  feet. 

12.  A  man  walks  from  A  to  B  in  h  hours.  If  he  had  walked 
a  miles  an  hour  faster  he  would  have  been  b  hours  less  on  the 
road.    Find  the  distance  from  A  to  B,  and  the  rate  of  walking. 

13.  Change  the  wording  of  12  to  suit  the  case  where  a  and  b 
both  change  signs. 


GEOMETRICAL    INTERPRETATIONS. 

106.  As  we  have  seen  in  Art.  21,  the  substitution  of 
numerical  quantity  for  the  quantitative  symbol  gives  an 
arithmetical  theorem  for  an  algebraic  expression,  and 
especially  for  an  identity. 

So,  also,  an  expression  written  in  the  symbolism  of 
algebra  may  admit  of  a  geometric  interpretation,  when 
the    quantitative    symbol    stands    for   some    elementary 


GEOMETRICAL   INTERPRETATIONS.  121 

geometric  idea.  The  mode  of  interpretation  must  depend 
upon  the  character  of  this  idea. 

By  employing  the  quantitative  symbol  to  stand  for 
different  geometric  ideas,  mathematicians  have  developed 
different  geometric  algebras,  requiring  different  modes 
of  interpretation,  and  some  of  which,  not  being  derived 
from  arithmetic,  are  not  subject  to  all  the  formal  laws 
deduced  in  Chapter  I.  It  need  scarcely  be  said  that 
these  latter  algebras  are  not  generalized  arithmetic,  and 
are  not  generally  applicable  to  numbers. 

Only  two  modes  of  interpretation  concern  us  here,  and 
these  are  such  as  belong  to  algebra  as  already  defined 
and  developed. 

The  distinction  between  these  is  as  follows: 

I.  The  quantitative  symbol  stands  for  a  given  portion 
of  a  straight  line,  or  a  line-segment,  as  a  geometric 
figure.  The  algebra  then  becomes  a  kind  of  symbolic 
geometry,  and  is  subject  to  certain  restrictions  arising 
from  the  nature  of  geometry. 

II.  The  quantitative  symbol  stands  for  a  number,  i.e. 
the  number  of  times  a  particular  line-segment,  taken  as 
a  unit  of  measure,  is  contained  in  a  given  line-segment. 

This  becomes  a  matter  of  concrete  quantity,  admitting 
of  geometric  interpretation,  and  being  subject  to  certain 
geometric  limitations. 

I.    Symbolic   Geometry. 

107.  The  primary  ideas  in  geometry  are  length  and  direc- 
tion, or,  mechanically  stated,  transference  and  rotation. 

Length  is  denoted  by  a  quantitative  symbol,  which  in 
this  connection  will  be  called   a  line-symbol.     Thus   a 


122  GEOMETRICAL   INTERPRETATIONS. 

denotes  a  given  line-segment.  But  it  does  more  than 
this ;  it  denotes  transference  from  one  end  of  the  seg- 
ment to  the  other  in  only  one  direction.  Then  —  a 
denotes  the  same  amount  of  transference  in  the  opposite 
direction  ;  and  thus  a  —  a  =  0  represents  a  neutralized 
effect,  and  is  equivalent  to  no  transference,  and  conse- 
quently to  no  line-segment. 

Thus  a  and  —  a  denote  the  same  segment  measured  in 
opposite  directions,  and  we  have  thus  a  simple  and  intel- 
ligible interpretation  of  the  signs  -f-  and  —  as  applied  to 
line-segments. 

Such  a  segment  is  called  a  directed  segment,  as  its 
direction,  being  one  of  two  opposites,  is  determined  by 
algebraic  sign. 

108.  a-\-b  is  a  segment  as  long  as  a  and  b  together; 
or  it  represents  transference  over  length  a  followed  by 
transference  over  length  b  in  the  same  direction. 

Thus,  if  a  =  AB,  b  =  BC,  g  ,,        b  . 

a  +  b  =  AB-{-BC=Aa  ^  b  c 

Then  a  —  &,  which  is  a  +  (  — 6)  represents  transference 
over  length  a  followed  by  transference  over  length  b  in 
the  opposite  direction.    If  a  is  longer  than  6,  let  a  =  AB, 

and   6=CjB.     Then    -b  =  BG,    ^^ , , , 

and  a  —  b  =  AB  +  BC,  or  trans- 
ference from  A  to  B,  followed  by  transference  from  B  to 
C.     This  is  equivalent  to  transference  from  A  to  (7,  and  is 
positive,  being  the  remainder  when  the  shorter  segment 
b  is  cut  off  from  the  longer  a. 

If  b  is  longer  than  a,  let  b  =  C'B. 

Then  a  —  6  =  AB  -f  BC  =  AC,  which  is  negative. 

Similarly,  na,  when  n  is  numerical,  denotes  a  segment 
n  times  as  long  as  that  denoted  by  a. 


GEOMETRICAL   INTERPRETATIONS.  123 

109.  The  area  of  a  rectangle  is  denoted,  in  the  sym- 
bolism of  algebra,  by  the  product  form  of  the  line- 
symbols  which  denote  two  adjacent  sides  of  the  rectangle. 

Thus  ah  means  the  area  of  the  rectangle  whose 
adjacent  sides  are  denoted  by  a  and  h. 

Then  a^  denotes  the  area  of  the  square  whose  side  is  a. 

If  either  a  or  6  is  negative,  the  area  ah  is  negative,  and 
is  subtractive  from  any  other  area  concerned.  Hence  an 
area  is  often  spoken  of  in  connection  with  the  symbolism 
of  algebra  as  a  directed  area.  A  square,  however,  is 
essentially  positive. 

110.  The  volume  of  a  rectangular  parallelopiped  or 
cuboid  is  symbolically  expressed  as  the  continued  product 
of  the  three  line-segments  which  denote  any  three 
conterminous  edges,  known  as  direction  edges,  of  the 
cuboid. 

Thus  ahc  is  the  volume  of  the  cuboid  having  a,  h,  and  c 
as  direction  edges. 

Then  a^,  which  is  the  same  as  aaa,  is  the  volume  of 
the  cube  whose  edge  is  a. 

We  thus  see  the  meaning  of  the  terms  square  and 
cube  as  introduced  from  geometry  into  arithmetical 
algebra. 

111.  The  evidence  of  the  legitimacy  of  the  conventions 
of  the  three  preceding  articles,  or  rather  the  proof  of  the 
necessity  of  109  and  110  as  following  from  the  funda- 
mental convention  of  107,  is  a  matter  for  geometry  rather 
than  for  algebra. 

In  the  elements  of  geometry  it  is  also  shown  that, 
with  these  conventions  in  regard  to  the  geometric  mean- 
ings of  the  algebraic  forms,  we  have 


124  GEOMETRICAL   INTERPRETATIONS. 

(1)  a +  6  =  6  + a. 

(2)  ab  =  ba,and  .*.  abc  =  acb  =  bac  =  etc. 

(3)  a{b -{- c)  =  ab -{- ac,  etc. 

Hence  the  symbols  a,  b,  c  are  subject  to  the  same 
formal  laws  of  transformation  whether  we  consider  them 
as  line-symbols  or  as  quantitative  symbols.  And  thus 
every  algebraic  identity  of  proper  form  may  be  inter- 
preted either  as  a  theorem  in  numbers,  i.e.  in  arithmetic, 
or  as  a  theorem  in  lines,  areas,  and  volumes,  i.e.  in 
geometry. 

EXERCISE  VIII.  b. 

Interpret  the  following  identities  as  geometric  theorems  — 

I.  (a-&)2  +  2a5  =  a2_^62. 

(a  —  6)2  is  the  square  on  the  difference  of  two  line-segments  ; 
ah  is  the  rectangle  on  the  segments  ;  and  a'^  +  &2  jg  ^he  sum  of  the 
squares  on  the  segments,  therefore 

The  square  on  the  difference  of  two  segments  and  twice  the 
rectangle  on  the  segments  are  together  equal  in  area  to  the  sum  of 
the  squares  on  the  segments. 

2.  a(a  +  b)=a^-\-ab.  5.    (a  +  6)2 -(«  -  6)2  .=  4a6. 

3.  (a+6)(a-6)=a2_52.      g.    ^a+by+(ia-by=2(a^-^b^y 

4.  (a  + 6)2  =  a2_,_  52  +  2  a6.      7.   ab(ia -}- b)=  a'^b -{- ab^ 

8.  (a  +  6  +  c)2  =  a2  +  52  +  c2  +  2(a6  +  6c  +  ca), 

9.  (a+by  =  a^-\-b^  +  Sab{a-\-b). 
10.  If  a  :  6  =  c  :  d,  then  ad  =  be. 

II.  If  a  :  6  =  6  :  c,  then  62  =  ac. 

12.   Ji  a:b  =  b;c  =  c:d,  then  a'.d  =  a^:bK 


GEOMETRICAL   INTERPRETATIONS.  125 

112.  Certain  restrictions  must  be  imposed  upon  alge- 
braic expressions  if  they  are  to  be  interpretable  as  real 
geometric  relations.  These,  besides  the  conditions  which 
render  an  expression  arithmetically  interpretable,  are  two 
in  number ;  namely, 

(1)  In  the  line-symbols  the  expression  must  be  of  not 
more  than  three  dimensions. 

This  is  due  to  the  fact  that  there  are  but  three  dimen- 
sions in  space,  the  subject-matter  of  geometry,  and  that 
by  our  convention  each  line-symbol  in  a  product  repre- 
sents one  of  these  dimensions. 

(2)  The  expression  must  be  homogeneous  in  the  line- 
symbols.  For  the  adding  of  one  species  of  magnitude  to 
another  species,  as  a  line  to  an  area,  or  an  area  to  a 
volume,  is  not  an  intelligible  operation. 

113.  Every  homogeneous  expression  of  one  dimension 
in  line-symbols  denotes  a  finite  line-segment,  or  is  linear. 

Every  homogeneous  expression  of  two  dimensions  in 
line-symbols  denotes  an  area;  such  areas  being  squares,  as 
a^,  or  rectangles,  as  ah,  or  areas  made  up  of  these. 

Every  homogeneous  expression  of  three  dimensions 
in  line-symbols  denotes  a  volume;  such  volumes  being 
cubes,  as  a^,  or  cuboids  on  square  bases,  as  a^h,  or  cuboids 
with  three  unequal  edges,  as  ahc,  or  volumes  made  up  of 
these. 

114.  An  expression  which  represents  a  geometric  rela- 
tion must  always  represent  a  geometric  relation  however 
it  is  transformed,  provided  it  is  interpretable.  And 
hence  a  homogeneous  expression  cannot  be  made  non- 
homogeneous  by  any  legitimit;^  tvmsformation. 


126  GEOMETRICAL   INTERPRETATIONS. 

This  fact  is  useful  in  many  ways. 

If  at  any  stage  in  the  transformations  of  algebraic  ex- 
pressions a  homogeneous  expression  becomes  non-homo- 
geneous, or  vice  versa,  some  error  in  work  is  to  be  looked 
for. 

Non-homogeneous    expressions   are    frequently   made 
homogeneous   in  form   by  the  introduction   of   a   unit- 
variable,  on  account  of  the  resulting  advantages  in  the  « 
after  work. 

Thus  0^24.  3a;— 2=0  may  be  written  x^+3xy -2y^=0, 
where  2/  is  a  variable  in  form  only,  and  is  to  be  replaced 
by  1  after  all  the  necessary  transformations  are  made. 

115.  In  applying  the  symbolism  of  algebra  to  develop 
metrical  relations  in  geometry,  a  sufficient  knowledge  of 
descriptive  geometry  is  required,  and  in  addition  to  the 
relations  already  laid  down  in  113  and  previous  articles, 
the  following  are  necessary : 

1.  If  a,  c  denote  the  sides  of  a  right-angled  triangle 
and  b  denote  the  hypothenuse,  6^  =  a^  -f  c^. 

2.  Similar  triangles  have  their  homologous  sides  pro- 
portional. 

Ex.  1.  Vab  is  linear,  and  denotes  the  side  of  the  square  whose 
area  is  equal  to  that  of  the  rectangle  whose  sides  are  a  and  b. 

Ex.  2.  Vabc  is  Hnear,  and  denotes  the  edge  of  the  cube  whose 
volume  is  equal  to  that  of  the  cuboid  of  which  a,  6,  c  denote 
direction  edges. 

Ex.  3.    Va^bc  is  an  area.    For  it  is  a  •  Vbc,  and  ■\/bc  is  linear. 

Ex.  4.  a^bCj  being  of  4  dimensions,  has  no  geometrical  inter- 
pretation. 


GEOMETRICAL   INTERPKETATIONS.  127 

Ex.  5.  ah  4-  he  is  the  sum  of  two  areas,  and  is  therefore  an  area ; 
but  ah  +  c  has  no  geometrical  meaning,  not  being  homogeneous. 

Ex.  6.  The  base  of  an  isosceles  triangle  is  h  and  its  altitude  is  a, 
to  find  the  perpendicular,  ^,  from  a  basal  vertex  to  a  side. 

Let  ABC  be  the  triangle  with  B  as  vertex,  D  the  foot  of  the  alti- 
tude, and  AE  the  required  perpendicular. 

Then  ah  =  AE .  BC,  since  each  expresses  double  the  area  of  the 
triangle. 

But  BDC  being  right-angled  at  Z>, 

J5C2  =  ^Z>2  _|.  2X72  =  a2  +  J  52. 
.\  ah=p^(ia^  +  ih^). 
Whence  p  r-        ^^^ 


V(4a^  +  62) 


Ex.  7.  If  any  given  area  be  divided  respectively  by  the  areas  of 
the  squares  on  the  two  sides  of  a  right-angled  triangle,  the  sum  of 
the  quotients  is  equal  to  that  obtained  by  dividing  the  given  area 
by  the  area  of  the  square  on  the  perpendicular  from  the  right  angle 
to  the  hypothenuse. 

Let  p  be  the  perpendicular. 

Then  ac=ph  =  twice  the  area  of  the  triangle. 

(Art.  115,  1) 


.-.  a-c2  = 

i)^62; 

=  p%a:^ 

+ 

C2) 

Whence 

1 

a2 

-1 

1 

-i)2> 

and  multiplying  by  any 

area, 

^2  say, 

W2 

a2 

^C2 

»2' 

which  interpreted  gives  the  theorem. 


128  GEOMETRICAL   LNTERPKETATIONS. 


EXERCISE  VIII.  c. 

1.  AA'  is  the  diagonal  of  a  square,  and  is  trisected  at  the  points 
C  and  D.     Find  the  area  of  the  square  when  the  segment  CD  is  a. 

2.  The  sides  of  a  rectangle  are  as  m  -.  n  and  the  diagonal  is  d  — 
i.   Find  the  sides. 

ii.   Find  the  area. 

iii.    Find  the  perpendicular  from  a  vertex  to  the  diagonal, 
iv.   Find  the  distance  between  the  feet  of  the  two  perpendiculars 
upon  the  same  diagonal. 

V.  Show  that  the  rectangle  on  the  diagonal  and  the  line-segment 
between  the  ieet  of  the  perpendiculars  on  the  diagonal,  is  equal  in 
area  to  the  rectangle  on  the  sum  and  difference  of  the  sides. 

3.  The  sides  of  a  rectangle  are  a  and  b  ;  to  find  — 
i.    The  perpendicular  upon  a  diagonal. 

ii.  The  distance  between  the  feet  of  the  perpendiculars  upon 
the  same  diagonal. 

iii.  Show  that  the  volume  of  the  cuboid,  whose  direction  edges 
are  the  two  sides  and  the  line-segment  of  ii.,is  equal  to  that  of  the 
cuboid,  whose  direction  edges  are  the  sum  and  difference  of  the 
sides  and  the  line- segment  of  i. 

4.  The  side  of  an  isosceles  triangle  is  n  times  the  altitude,  and 
the  base  is  2  &  ;  to  find  the  area.  What  does  the  result  become  when 
n=  1?    Explain. 

6.  The  base  of  an  isosceles  triangle  is  one-half  the  side,  and  the 
perpendicular  upon  the  base  is  J  Vl5.     Find  the  area. 

6.  An  upright  tree  is  broken  over  by  the  wind,  and  the  top 
touches  the  ground  at  36  feet  from  the  base  Find  the  length  of 
the  whole  tree  when  the  remaining  upright  part  is  15  feet. 

7.  How  large  a  circular  disc  can  be  cut  from  a  triangular  piece 
of  paper  whose  edges  are  13,  14,  and  15  inches  respectively  ? 


GEOMETRICAL   INTERPRET ATIONS.  129 

8.  In  a  right-angled  triangle  the  median  to  the  hypothenuse  is 
r  times  one  of  the  sides  ;  find  the  ratio  of  the  sides  to  one  another. 

9.  In  an  isosceles  triangle  where  h  is  the  base  and  s  the  side, 
the  area  is  expressed  by  —  ;  find  the  ratio  of  the  side  to  the 
base.  ^ 

10.  If  the  side  of  a  square  be  increased  by  -th  of  itself,  where  n 

n 
is  a  large  number,  by  what  part  of  itself  is  the  area  increased  ? 

11.  If  the  edge  of  a  cube  be  increased  by  -th  part  of  itself, 

n 
where  n  is  a  large  number,  by  what  part  of  itself  is  the  volume 
increased  ? 

12.  Find  the  ratio  of  the  diagonal  of  a  square  to  — 

i.   The  side, 
ii.   The  join  of  a  vertex  with  the  middle  of  a  side. 

13.  Find  the  ratio  of  the  diagonal  of  a  cube  to  its  edge. 

14.  Compare  the  area  of  an  equilateral  triangle  on  the  side  of  a 
square  to  one  on  the  diagonal. 

15.  A  boat  making  10  miles  an  hour  in  still  water  steers  directly 
across  a  stream  flowing  4  miles  an  hour.  Compare  the  real  velocity 
of  the  boat  with  — 

i.   Its  velocity  across  the  stream. 

ii.    Its  velocity  down  the  stream. 

16.  A  boat  goes  a  certain  distance  down  a  stream  in  t  seconds, 
and  requires  t^  seconds  to  return.  Compare  the  velocity  of  the 
boat  with  that  of  the  stream. 

Interpret  when  t  ^t^. 

17.  A  street  60  ft.  wide  has  a  house  20  ft.  high  on  one  side, 
and  a  house  30  ft.  high  on  the  other.  How  long  a  ladder  is 
required,  and  where  must  its  foot  be  placed,  that  it  may  just  reach 
to  the  top  of  each  house  ? 


130 


THE   GKAPH. 


Y 


II.   Geometry  as  Concrete  Quantity.  —  The  Graph. 

116.    Take  any  point  0,  and  through  it  draw  the  two 
lines  XX\  YY'  at  right  angles  to  one  another. 

These  lines  are  lines  of 

^  reference,  and  are  called 

axes,  and  0  is  the  origin. 

To  distinguish  the  axes, 

XX'  is  called  the  avaxis, 

X     and  YY^  the  y-axis. 

Measures  are  made 
along  the  ar-axis  from  O 
to  the  right  or  left,  those 
to  the  right  being  posi- 
tive and  those  to  the  left  being  negative.  Also  measures 
are  made  from  the  aj-axis  parallel  to  the  ^/-axis,  upwards 
or  downwards,  those  taken  upwards  being  positive  and 
those  downwards  negative.  We  have  thus  two  sets  of 
measures  which  represent  the  two  dimensions  of  the 
plane,  and  by  means  of  the  convention  of  signs  stated 
above  we  may  represent  any  point  in  the  plane. 

Let  P  be  any  point,  and  FM  be  perpendicular  to  OM. 
Then  the  measures  which  determine  the  position  of  P 
relatively  to  the  origin  and  axis  are  OM  and  MP^  and 
if  these  are  known,  the  position  of  P  is  known. 

Usually,  and  for  the  sake  of  uniformity,  the  measure 

OM  IS  called  the  x  of  the  point  P,  and  the  measure  MP 

is  called  the  y  of  the  point  P,     If,  then,  the  x  and  y  of 

a  point  are  given,  the  point  can  be  laid  down. 

Thus,  let  the  a;  of  P  be  2  and  its  y  be  3. 

To  get  the  point,  we  measure  OM  to  the  right  equal 


THE  GKAPH.  131 

to  2  units  from  any  adopted  scale,  and  then  measure  MP 
upwards  parallel  to  OF  and  equal  to  3  units  from  the 
same  scale.  The  point  thus  found  is  an  ocular  represen- 
tation of  the  given  point,  and  is  called  the  graph  of  the 
given  point. 

If  X  were  ~  2,  and  y  3,  we  would  take  OM^  to  the 
left  and  get  the  point  Pi ;  and  if  the  point  had  its 
a;=  —  2  and  its  y  =  —  3,  we  would  also  take  J/1P2  down- 
ward and  get  the  point  Pg-  Thus  any  point  is  completely 
determined  by  its  x  and  y  with  their  proper  signs. 

117.  Now  let  y=fx  be  any  integral  function  of  x. 
For  every  value  of  x  we  have  a  corresponding  value  of  y, 
and  if  x  varies  continuously,  i.e.  by  infinitely  small 
gradations,  y  also  varies  continuously. 

Let,  then,  a  number  of  corresponding  values  of  x  and 
y  be  found  by  giving  to  x  any  convenient  arbitrary 
values,  and  finding  the  resulting  value  of  y  for  each. 
These  values  form  the  x's  and  y^s  for  a  set  of  points 
whose  graphs  all  lie  upon  the  graph  of  the  function  /ic, 
this  graph  being  a  line  or  curve  passing  through  all  the 
points. 

If  all  possible  values  of  x  could  be  considered,  the 
points  would  be  infinite  in  number,  and  would  exactly 
mark  out  the  graph  of  the  function ;  but  as  we  cannot 
practically  take  every  value  of  x,  we  take  a  set  of  values, 
usually  integral,  as  being  most  convenient,  and  thus  get 
a  set  of  points.  We  then  connect  these,  as  well  as 
possible,  by  a  line  or  curve,  as  may  be  required. 

The  theoretical  graph  is  an  exact  geometrical  picture 
of  the  function,  and  in  itself  and  in  its  relation  to  the 
axis  represents   every   property  of  the  function.     The 


132 


THE   GRAPH. 


practical   graph,  is  a  more  or  less  close  approximation 
to  this. 


Ex.  1.  Let 

y  =  2x-l. 

Take 

x  =  -l,      0,  +  l,  +  2, 

Then 

2/ =  -3,  -1,  +  1, +3, 

c/ 


-X- 


Lay  down  the  graphs  of  the  points  whose  a;'s  and  i/'s 

are  given,  as  at  a,  h,  c,  d,  "-  in  the  diagram. 

It  is  readily   seen   that 
^  /  ... 

these  points  lie  in  line,  and 

that  the  graph,  which  is 
denoted  by  the  dotted  line, 
is  a  straight  line.  Hence 
the  reason  for  calling  2x  —  l 
a  linear  function  of  x,  and 
^  2a;— 1  =  0  a  linear  equa- 
tion. 

It  can  be  readily  shown 
that  the  graph  of  every 
function  of  the  form  ax-\-h 
is  a  straight  line. 

The  root .  At  the  point 
R,  where  the  graph  ad  cuts 
the  ic-axis,  we  have  y  =  0, 
and  .*.  2x^1  =  0.  The  x  of  this  point  is  OB^^^nd  as 
this  is  the  value  of  x,  which  makes  the  function  zero, 
OE  measures  the  value  of  the  root.  It  is  readily  seen 
that  OiiJ  =  f 

Thus  the  cutting  of  the  ic-axis  by  the  graph  denotes  a 
iral  root,  and  the  distance  from  the  origin  to  the  point 
of  intersection  measures  the  value  of  the  root,  +  if  to 


/ 


/ 


J 


THE   GRAPH. 


133 


the  right,  and  —  if  to  the  left.     In  the  present  example 
the  root  is  -f . 

As  the  line  ad  must  cut  the  a^axis  either  at  a  finite 
point  or  at  go,  a  linear  equation  has  one  root  only ;  this 
root  must  be  real,  and  may  be  finite  or  infinite  in  value. 

Ex.  2.  Let  y  =  a^  +  x^-2x-l  =fx. 

Take  a;  =  -2-l      0  +  1  +  2. 

Then  ?/  =  -l  +  l-l-l  +  7. 

The  graph  is  given  at  G  in  the  diagram,  being  a  curve 
through   the  points 
a,  6,  c,  d,  e  •••  / 

1.  The  graph  cuts 
the   a.'-axis  at  three 
points,   Bj    E',   and 
E". 

Hence  there  are 
three  real  roots  to 
the  equation 
a;3_^^.2_2a;-l  =  0. 
.  Two  of  these,  OE 
and  OE',  are  nega- 
tive, and  the  third, 
OE'',  is  positive. 
The  limits  of  the  roots  are,  OE  >  —  2  and  <  —  1,  OE' 
>  -  1  and  <  0,  and  OE"  >  1  and  <  2. 

2.  If  we  move  the  graph  bodily  upwards  through  1 
unit,  we  bring  it  to  the  position  G',  and  increase  every 
value  of  y  by  1  unit.  But  we  may  increase  the  value  of 
2/  by  a  unit,  by  adding  a  unit  to  the  independent  term  of 
the  function. 


I 

'''I 

e=7 

' ' 

^\ 

/' 

f       \ 

' ; ! 

/            \ 

/ 1 ' 

k    \ 

/   1 

/  / 1 

'^v 

^\                /  / 1 

1       t       \ 

\                '  1 

1   '     ^ 

\       / ' 

/     /G'    6    \ 

\     ''  / 

''  /  /I 

\   ^ 

/  /  r 

\ 

/ 

/ 
/ 

/'R    -1     R'^ 

/ 

/ 

\ 

1    /R''2          X 

/ 

/ 

/ 

a 

c 

\        y 

d 

I 


134  THE  GRAPH. 

Hence,  to  increase  the  independent  term  of  the  func- 
tion is  equivalent  to  moving  the  graph  upwards,  and  to 
decrease  it  is  equivalent  to  moving  the  graph  downwards. 

(7'  is  thus  the  graph  of 

in  which  a  has  come  to  —  2,  c  to  0,  and  d  to  1. 

The  roots  are  now  —  2,  0,  and  1. 

The  two  points  E'  and  E"  have  come  nearer  together, 
and  the  two  E  and  E'  have  gone  further  apart ;  that  is,  the 
roots  OE*  and  OE'^  have  approached  one  another  in  value, 
while  the  roots  0^'  and  OE  have  become  farther  sepa- 
rated in  value. 

3.  Add  another  unit  to  the  independent  term,  and  the 
graph  is  moved  into  the  position  (^",  which  is  the  graph 

of  a^  4-0:2  _2aj  +  l. 

As  the  loop  L  no  longer  cuts  the  aj-axis,  the  points  E' 
and  E"  have  become  imaginary,  while  the  point  E  is 
still  real.  Hence  the  two  roots  0^'  and  0^"  of  G  have 
become  imaginary  in  (r",  while  the  third  root,  OE, 
remains  real.  Thus,  then,  o^  +  iB^  — 2aj  +  l  has  two 
imaginary  factors,  i.e.  complex  quantities,  and  one  real 
factor. 

4.  In  the  motion  of  the  graph  from  the  position  G'  to 
that  of  6r''  there  was  an  intermediate  position  in  which 
the  loop  L  just  touched  the  ic-axis.  The  points  E'  and 
E"  were  then  coincident,  and  the  roots  OE'  and  OE'' 
were  identical. 

We  thus  see  that  in  passing  from  real  to  imaginary 
two  roots  approach  one  another  in  value,  become  equal, 
and  then  become  imaginary ;  and  since  two  roots  must 
always  be  thus  involved  together,  the  roots  must  become 


THE   GRAPH.  135 

imaginary  in  pairs ;  or,  more  concisely,  imaginary  roots 
exist  in  pairs. 

5.  The  least  consideration  will  show  that  similar 
changes  take  place  when  the  graph  is  lowered  by  sub- 
tracting from  the  independent  term  of  the  function,  the 
difference  being  that  B  and  R^  will  then  become  imag- 
inary, while  i?"  remains  real. 

The  mode  of  representing  a  function  by  a  graph  is 
due  to  Descartes,  and  its  invention  is  one  of  the  great 
milestones  in  the  progress  of  mathematics.  The  graph 
is  largely  employed  by  statisticians,  by  engineers,  by 
physicists,  by  chemists,  and  many  others  who  are  able 
to  employ  mathematical  methods  intelligently;  and  its 
systematic  discussion  is  the  subject-matter  of  coordinate 
geometry. 

EXERCISE  Vni.  d. 

1.  Construct  the  graphs  of  2  ?/  +  3  a;  =  6,  and  of  3  y  —  2  «  =  6. 

2.  Construct  the  graph  of  x  —  y  =  0.  How  is  it  situated  with 
respect  to  the  axes  ? 

3.  In  the  graph  of  ax  -\-  hy  -{■  c  =  0,  what  is  the  effect  of  — 

i.  Increasing  the  independent  term  ? 
ii.   Increasing  the  coefficient  of  x  ? 
iii.   Increasing  the  coefficient  of  ?/  ? 

4.  Draw  the  graphs  of  ic^  —  4^  +  2  =  2/;  oi  x^  +  ^x"^  —  x  —  \  =  y] 
and  oix^  —  4:X  =  y.  How  are  these  graphs  situated  in  relation  to 
one  another  ? 

6.  What  integer  added  to  the  independent  term  of  a^  —  2  ic^  —  x^ 
+  2  X  —  1  will  make  all  the  roots  imaginary  ?  Will  make  all  the 
roots  real  ? 

6.  Explain  from  the  graph  why  a  cubic  must  have  one  re^  xoot. 


CHAPTER   IX. 

The  Quadratic. 

118.   The  most  general  type  of  a  quadratic  function  of 
one  variable  is 

ax^  -{-hx  -{-c, 

and  the  corresponding  equation  is 

ax^-\-'bx-\-c  =  0 {A) 

In  the  equation  we  may  divide  through  by  a ;  then 

a        a 

b  c 

and  writings  for  -,  and  q  for  -,  the  equation  becomes 
a  a 

x^-}-px  +  q  =  0] (B) 

which  is  the  quadratic  reduced  to  its  simplest  foi-m.     The 
roots  of  this  are,  by  Art.  59, 


i«i  =  i(  — p+Vp^  — 4g),  and  0^2  =  i(  — P  — Vp^  —  4g). 

On  account  of  the  double  sign  of  the  root-symbol,  -yj 
(Art.  48),  both  values  are  included  in  the  one  expression 

^  =  i(— i>  ±yp^  — 4g), 

and  this  is  the  solution  of  {B), 
136 


THE  QUADRATIC.  137 

h  c 

In  this  solution  write  -  for  p,  and  -  for  q,  and  re- 
duce, and  we  obtain 

Jj  (I 

and  this  is  the  solution  of  (A) 

The  forms  of  these  solutions  should  be  so  mastered 
that  for  any  quadratic  equation,  in  either  of  the  forms 
(A)  or  (B)y  the  solution  may  be  written  down  at  once. 

Ex.  1.   The  roots  of  3  a:2  +  2  x  -  4  =  0  are 


x  =  K-2±V4T48)=K-l±V13). 
Ex.  2.   The  roots  of  2a:2  -  Sx  +  2  =  0  are 

X  =  i(S±V9^=^)=l(S  ±  i  VT). 

119.  The  double  root,  or  double  solution  of  the  quad- 
ratic, is  frequently  of  the  highest  importance  as  giving 
an  unexpected  answer  to  a  problem,  and  through  this 
answer  giving  us  a  clearer  idea  of  the  nature  of  the 
problem. 

It  is  only  when  a  problem  admits,  in  spirit,  of  a 
double  answer,  that  it  involves  the  solution  of  a  quadratic. 

A  few  examples  will  make  this  plain. 

Ex.  1.  A  man  buys  a  horse  and  sells  him  for  $24,  thus  losing  as 
much  per  cent  as  the  horse  cost  in  dollars.     To  find  the  cost. 

X 

100* 

Or  x2  -  100  x  + 2400  =  0. 

Whence  aj  =  60  or  40. 

This  solution  shows  the  problem  to  be  to  a  certain  extent 
indefinite,  since  there  is  no  way  of  determining  whether  the  cost 
of  the  horse  was  $40  or  $60. 


138  THE   QUADRATIC. 

Ex.  2.  The  attraction  of  a  planet  varies  directly  as  its  mass  and 
inversely  as  the  square  of  the  distance  from  its  centre.  The  earth's 
mass  is  75  times  that  of  the  moon,  and  their  distance  apart  is 
240000  miles.  To  find  a  point,  in  the  line  joining  them,  where  their 
attractions  are  equal. 

Let  P  be  the  point,  and  let    0 , ^ , 

EP=X.  E  P  M  Q 

Attraction  ofE  =  — ;  and  of  If  = :  and  these  are 

x-2  (240000  -  xy^ 

to  be  equal.    This  gives 

74  x2  _  150  X  240000  a;  +  76  •  (240000)2  =  0. 

Whence  x  =  215160  or  271330  miles. 

The  smaller  of  these  numbers  evidently  gives  EP ;  the 
larger,  being  greater  than  240000,  gives  a  second  point, 
Q,  beyond  the  moon,  and  not  contemplated  in  the  prob- 
lem.    Our  judgment  tells  us  that  there  is  a  second  point. 

Cor.  In  the  foregoing  question  let  the  masses  of  the  moon  and 
earth  be  the  same.     Then  we  have 

or  ^2(1  -  1)  -  480000  x  +  (240000)2  =  0. 


x^      (240000  -  x) 

By  Art.  77,  x  =  00  or  120000. 

That  is,  one  point,  P,  is  half  way  between  the  earth  and  moon, 
and  the  other  is  infinitely  distant. 

EXERCISE  IX.  a. 

1.    Solve  the  quadratics. 

I.  x^  +  x-i(b-c)-ibc  =  0.       iv.  3a;2__2a;+l  =  0. 

II.  x^-ax+  i(a^  -  62)  =  0.  v.   (a2  -  b^)x'^  -2ax  +  l=0. 
ill.  abx^  -  (a2  +  b^)x  +  a6  =  0.        vi.  x2  -  x  -  J  =  0. 


THE   QUADRATIC.  139 

2.  Find  the  relation  between  a  and  h  in  the  equation 
(a  -\-  x)  (h  —  x) -\-  abx  —  1=0,  when  — 

i.    The  sum  of  the  roots  is  zero, 
ii.   The  sum  of  the  reciprocals  of  the  roots  is  zero, 
iii.    The  sum  of  the  reciprocals  of  the  roots  is  infinite. 

3.  If  the  equation  ax^  +  bx  +  c  =  0  has  a  and  p  as  its  roots, 

find  the  equation  which  has  -  and  -  as  its  roots. 

/3 

4.  Show  that  the  roots  of  ax^  +  bx+  a  =  0  are  reciprocals  of 
one  another. 

5.  The  area  of  a  right-angled  triangle  is  a^  and  the  difference 
between  the  two  sides  is  d;  to  find  the  sides.  Explain  the  double 
solution,  and  draw  figures  to  represent  it,  when  a^  —  4  and  d  =  2. 

6.  ABCD  is  a  square.  P  is  a  point  on  AB  produced,  and  Q  is 
on  AD^  and  PCQ  is  a  right  angle.  Determine  BP  so  that  the 
triangle  PCQ  shall  have  a  given  area,  a^.  Explain  the  double 
solution. 

7.  In  Ex.  6,  ^§  is  equal  to  BP\  determine  BP  when  the  tri- 
angle P  CQ  has  a  given  area,  a^.    Explain  the  double  solution. 

8.  AB  and  CD  are  two  straight  lines  intersecting  at  right  angles 
in  0.  AC  is  of  a  given  length,  I.  Find  AO  when  the  triangle 
ACQ  has  a  given  area,  a"^.     Explain  the  quadruple  solution. 

9.  Find  the  area  of  the  triangle  of  Ex.  8,  when  AO  =  2  CO, 
10.    Find  CO,  of  Ex.  8,  when  AO'^  =  l'CO. 

120.    The  rational  part, ,  in  the  solution  of  (A) 

Jj  a 

is  the  same  for  each  root,  the  difference  in  the  roots 
being  due  to  the  part  V^^  —  4  ac. 

As  this  part  may  be  rational,  irrational,  or  imaginary, 
both  roots  are  alike  rational,  irrational,  or  imaginary. 

(1)  When  V&^  — 4:ac  is  real,  the  roots  are  real  and 
different. 


140 


GKAPH    OF   QUADRATIC. 


This  occurs  when  a  and  c  have  unlike  signs,  or  when 
they  have  like  signs  and  6^  >  4  ac. 

Ex.  1.    The  roots  of  x^  -  2  x  -  2  are  1  di  V'*^- 
Ex.  2.    The  roots  of  x^  -  3  x  +  1  are  3  ±  V^- 

(2)  When  -y/b^  —  4  ac  =  0,  the  roots  are  real  and  equal. 
This  occurs  when  &^  =  4  ac,  in  which  case  the  function 

is  a  complete  square. 

Ex.  3.    The  roots  of  ic2  -  4  x  +  4  are  2  ±  0. 

(3)  When  V&^  — 4ac  is  imaginary,  the  roots  are  com- 
plex numbers,  unless  h  is  zero,  when  they  are  imaginaries. 

This  occurs  w^hen  6-  <  4  ac. 


Ex.  4.    The  roots  of  x^  -  2  ic  +  2  are  1  ±  i. 


± — V— 4ac,   and 
2a 


(4)  When   6  =  0,    the   roots    are 

differ  in  sign  only ;  but  they  may  be  rational,  irrational, 
or  imaginary. 

(5)  If  the  roots  are  real,  and  a  is  +,  they  will  have 
the  same  sign  when  b  >  V&^  —  ^ac\  that  is,  when  c  is  +. 
The  sign  of  the  roots  will  be  the  opposite  to  that  of  b. 

This  takes  place  in  Ex.  2,  the  roots  being  real,  and  a 
and  c  being  both  +. 

121.  The  Graph.  The 
graph,  G,  of  x^—^x-\-l  is 
given  in  the  margin.  The 
roots  OR  and  Oi^'  are  both 
positive  (Art.  120,  5). 

Let  x^-{-px-{-q=0  be  the 
quadratic  in  form  {B),  and 
let  Xi  and  X2  be  the  roots. 


MINIMUM   AND  MAXIMUM.  141 

Then, 

(X  —  Xi)  {X  —  X^  =  X^  —  {Xi  -f-  ^2)  ^  +  ^1^2  =  ^ 

is  the  equation  ;  and  comparing  with  the  former,  we 
have  Xy  +  .^2  =  —  p,  and  x^x^  =  g.     That  is  — 

The  sum  of  the  roots  is  the  coefficient  of  linear  x  with 
changed  sign,  and  the  product  of  the  roots  is  the  inde- 
pendent term. 

If  we  put  for  p  and  q  their  vahies  in  terms  of  a,  6,  and 

c,  we  get  Xi-{-X2  — ,  and  X1X2  =  -• 

a  a 

Since  x^  +  X2  is  independent  of  g,  the  sum  of  the  roots 
is  not  affected  by  changing  the  value  of  g.  Hence  if  we 
move  the  graph  upwards  by  adding  to  g  (Art.  117,  Ex.  2, 
2)  until  L  comes  to  (7,  the  roots  become  equal  and  their 
sum  is  unchanged.     Hence  0(7=  ^  (OjR  -f-  Oi?'). 

122.   Minimum  and  Maximum. 

The  y  of  any  point  of  the  graph  expresses  the  value 
of  the  function  a,^  —  3  a?  +  1  for  the  corresponding  value 
of  X ;  thus  for  a;  =  0  the  value  of  the  function  is  Oa,  for 
a;  =  01  the  value  is  16,  and  for  x  =  OG  the  value  of  the 
function  is  CL. 

The  function  has  then  a  least  value  (7X,  called  its 
minimum,  but  it  has  no  greatest  value. 

If  we  change  the  signs  of  the  function  throughout,  we 
do  not  affect  the  roots  in  any  way,  but  we  change  the 
sign  of  every  value  of  y,  and  we  thus  reverse  the  graph, 
putting  it  into  the  position  g. 

The  function  now  has  a  greatest  value  CL\  its  maximum, 
but  it  has  no  least  value. 

Hence  a  quadratic  function  with  the  coefficient  of  x^ 
positive  has  a  minimum  value  but  no  maximum ;  and 


142  MINIMUM   AND   MAXIMUM. 

with  the  coefficient  of  os?  negative,  it  has  a  maximum,  and 
no  minimum. 

123.    To  find  the  minimum  or  maximum  solution. 

It  appears,  from  Art.  121,  that  when  ic  =  OC,  the  value 
of  the  function  is  either  a  minimum  or  a  maximum, 
according  as  the  coefficient  of  o?  is  positive  or  negative. 

But  0(7=  1  (0/?  +  OjK')  =  one-half  the  sum  of  the  roots 
=  — ^p.  Hence  the  required  solution  is  obtained  by 
substituting  for  x  one-half  the  coefficient  of  linear  x  with 
changed  sign. 

Ex.  1.  The  minimum  value  of  x2  -  3x  +  1  is  (|)2  -  3(|)+  1, 
or  -  f  =  (7/:. 

Ex.  2.    To  divide  a  number  into  two  parts   such  that  their 
product  may  be  a  minimum  or  a  maximum,  and  to  find  its  value. 
Let  a  be  the  number,  and  x  one  of  the  parts. 
Then  x{a  —  x)  is  to  be  a  minimum  or  a  maximum. 
But  the  function  ax  —  x^  has  a  maximum  solution  (Art.  122). 
-,  and  X  is  -,  or  the  number  is  halved. 


laY_a^ 


2 


Ex.  3.  Two  trains  A  and  B  are  on  two  roads  crossing  at  right 
angles  and  approaching  the  crossing.  A  is  a  miles  from  the  cross- 
ing and  goes  a  miles  an  hour  ;  B  is  5  miles  from  the  crossing  and 
goes  /3  miles  an  hour.  When  will  they  be  nearest  together,  and 
how  far  apart  will  they  then  be  ? 

Let  X  be  the  time  in  hours.    Then — 

a  —  ax  is  A' s  distance  from  the  crossing  at  the  end  of  x  hours, 
and  6  —  j3x  is  B's  distance. 

Their  distance  apart  is  V(a— ax)^  +  (5— jSx)^,  and  this  is  to  be 
a  minimum.     But  its  square  will  also  be  a  minimum. 

.-.  (a-ax)2  +  (6-/3x)2 
or  x\a^  4-/32)  -  2  a;(aa  +  &3)  -f  «2  +  62 

is  to  be  a  minimum. 


MINIMUM   AND  MAXIMUM.  143 

The  value  of  x  is  «fL±_M. 

a'  +  )82 

If  this  value  be  substituted  for  x,  the  function  reduces  to 

{hoL  -  a&y 
a2  +  )82    ' 

which  is  the  square  of  the  least  distance. 

124.    We  arrive  at  the  results  of  Art.  123,  without 
using  the  graph,  as  follows : 

Let  x^+px  +  q  =  y. 


Then  a;  =  i(  — p  ±  Vp^  —  4g  + 4y). 

Now,  whatever  be  the  value  of  jp^  — 4g,  the  expression 
Vi>^  — 4g  -f  4?/  cannot  be  made  imaginary  by  increasing 
the  value  of  y,  while  it  may  be  made  so  by  sufficiently 
diminishing  the  value  of  y.  If,  then,  the  roots  are  to  be 
real,  y  has  a  minimum  value,  and  this  minimum  is 
reached  just  as  the  expression  p^  —  4:q  -{-  4,y  is  passing 
from  +  to  — ;  i.e.  when  the  expression  is  zero. 

This  gives  x  =  —  ^p  for  the  minimum  solution ;  and 
the  value  of  y  is  found  either  by  substituting  this  value 
of  X  in  the  function,  or  by  putting  p^  —  Aq  -\-  4:y  equal  to 
zero  and  solving  for  y. 

Hence  y  =  —  ^  (p^  —  4g)  ;  i.e.  one-fourth  of  the  quan- 
tity under  the  sign  ^,  in  the  solution  of  the  equation 
oc^  +  px  -{-  q  =:  0,  with  its  sign  changed. 

Next  let  —  x^  -\-  px  -\-  q  z=  y. 

Then  x=:^(p  ±  Vp^  -}-  4g  —  4:y). 

The  part  Vp^  +  4g  — 4y  may  be  made  imaginary  by 
increasing  the  value  of  y,  but  not  by  diminishing  it. 
Hence  the  function  now  admits  of  a  maximum  value, 


144  THE   QUADRATIC. 

but  not  of  a  minimum  ;  and  the  maximum  solution  as 
before  is  given  hj  x  =  ^p,  and  the  value  of  the  maximum 
is  2/  =  i(P^  +  4g). 

As  the  value  of  x,  which  gives  a  maximum  or  a  min- 
imum solution,  does  not  involve  the  irrational  part  of  the 
root,  the  solution  is  independent  of  the  nature  of  the 
roots,  as  to  whether  they  are  real  or  imaginary. 

125.  By  studying  the  graph,  we  see  that  for  real  roots 
with  x^  positive,  the  function  has  a  negative  value  for 
all  values  of  x  lying  between  the  roots,  and  positive  for 
values  lying  beyond  the  roots ;  and  for  a^  negative,  the 
value  of  the  function  is  positive  for  all  values  of  x  lying 
between  the  roots,  and  negative  for  all  values  of .  x  lying 
beyond  the  roots. 

Ex.   For  what  values  of  5c  is  3  ic^  —  2  ic  —  1  positive  ? 
The  roots  of  3  a;^  —  2  x  —  1  =  0  are  1  and  —  i  ;  and  the  expres- 
sion is  positive  for  every  value  of  a;  >  1  and  <  —  |. 

And  the  expression  is  negative  for  every  value  of  x  <  1  and 

EXERCISE  IX.  b. 

1.  Construct  the  graphs  of  — 

i.    x^  —  X—  1.  iii.   2  +  X  —  x^. 

ii.   ^2  +  X  +  1.  iv.   2  —  X  —  x^. 

2.  Construct  the  graph  of  4  x^  +  4  x  +  1. 


3.    Construct  the  graph  of  \/4  —  x^. 

Here  we  put  y  =  \/4  —  x'^,  and  hence   2/^  =  4  —  x^  ;   and  y  has 
thus  two  values  differing  in  sign  only  for  every  value  of  x. 


4.  Construct  the  graph  of  \/4x. 

5.  Construct  the  graph  of  x(l  ±  2). 


THE  QUADRATIC.  145 

6.  Construct  the  graph  of  x^  +  ic  +  1.  •     . 

7.  Construct  the  graph  ot  x^—  x"^  —  x  +  1. 

8.  Construct  the  graph  of  ^^  "  ^  +  ^  . 

x-{- 1 

9.  Find  the  maximum  or  minimum  value  of  the  following 
functions : 

i.  x^  +  x-1.  iii.  3x-x2  +  2. 

ii.  3  x2  -  2  a;  -  1.  iv.  x'^-Sx. 

10.  Find  the  numerical  quantity  which  exceeds  its  square  by 
the  greatest  possible  quantity. 

11.  Divide  a  number  into  two  parts  such  that  the  sum  of  the 
squares  of  the  parts  may  be  a  minimum. 

12.  Find  the  number  which  when  added  to  its  reciprocal  gives 
the  smallest  sum. 

13.  Divide  a  number  a  into  two  parts  such  that  the  square  of 
one  part  added  to  n  times  the  square  of  the  other  may  be  the  least 
possible,  and  find  the  sum. 

14.  Divide  a  number  into  two  parts  such  that  the  difference 
between  the  sum  of  the  squares  upon  the  parts  and  the  product  of 
the  parts  may  be  a  minimum. 

15.  Divide  20  into  two  parts  such  that  their  product  may  be  120. 
The  result   is   x  =  lO  ±2i  ^^.      The  factors   2(5  +  iy/5)  and 

2(5  —  ^V5)  have  20  as  their  sum  and  120  as  their  product,  and 
thus  algebraically  the  problem  is  solved.  But  the  complex  num- 
bers tell  us,  in  the  only  way  in  which  algebra  can  do  so,  that  the 
question  is  arithmetically  absurd  or  impossible.  We  are  shown 
why  this  is  so  in  Ex.  2  of  Art.  123. 

16.  The  sum  of  a  quantity  and  three  times  its  reciprocal  is  y'S  ; 
is  the  quantity  real  or  imaginary  ? 

17.  Show  that  -^ cannot  be  greater  than  f  ^3  if  x  is 

real,  x^-x  +  1 


146  THE  QUADRATIC. 

x^  A-  or 1 

18.  If  A:  is  a  value  of  x  which  makes  — — equal  to  2, 

x^  —  X  -i-  1 
show  that  A:  is  a  complex  number.  ^ 

19.  ABCD  is  a  square  ;  on  AD  a  point  P  is  taken,  and  on  AB 
a  point  §,  so  that  AP  =  BQ.  Find  AP  when  the  area  of  the  tri- 
angle QAP  is  a  given  quantity,  a^. 

Denote  the  side  of  the  square  by  s,  and 
let  AP  =  BQ  =  X.  Then  AQ  =  s  -  x,  and 
the  area  of  the  triangle  APQ  is 

J  x(s  —  x)=  a^. 


Thence,    x  =  i(s±  Vs^  -  8 a^) . 

(1)  There  are  two  solutions,  and  there- 
fore two  positions  for  P.  This  is  seen  in 
the  diagram,  in  the  triangle  AP'Q'. 

(2)  a^  has  a  maximum,  that  is  when  8  a^  —  s2^  or  the  area  of  the 
triangle  is  one- eighth  that  of  the  square. 

(3)  When  the  triangle  has  its  maximum,  the  two  solutions 
become  one,  and  x  =  J  s,  as  is  seen  in  the  triangle  AP"Q". 

20.  In  Ex.  19,  Q  is  taken  on  AB  produced,  so  that  BQ  =  AP, 
Examine  the  case  and  show,  (1)  that  there  are  two  solutions  for  a 
given  area  of  triangle,  (2)  that  the  triangle  has  a  minimum,  and 
(3)  that  for  the  minimum  x  =  —  J  s,  and  a^  =  —  ^  s^,  and  explain 
these  negative  quantities. 

21.  Examine  Ex.  19,  when  BQ  is  so  taken  that  the  rectangle 
contained  by  AP  and  BQis  2i  constant,  c^. 

22.  In  Ex.  6,  of  IX.  a,  has  the  triangle  PQC  a  maximum  or  a 
minimum,  and  what  is  its  value  ? 

23.  In  Ex.  7,  of  IX.  a,  has  the  triangle  PQG  a  maximum  or  a 
minimum,  and  what  is  its  value  ? 

24.  Eind  the  maximum  value  of  the  triangle  AOC  in  Ex.  8  of 
IX.  a. 

25.  A  rectangular  field  is  to  contain  an  acre  of  ground,  and  a 
path  from  one  comer  to  the  middle  of  an  opposite  side  is  to  be  as 


THE   QUADRATIC.  147 

short  as  possible.    What  must  be  the  form  and  dimensions  of  the 
field? 

26.  An  isosceles  triangle  has  its  equal  sides  given,  to  find  the 
third  side  when  the  area  is  a  maximum. 

27.  Along  a  road  already  fenced  a  rectangular  plot  of  1  acre  is 
to  be  inclosed.  What  must  be  its  form  that  the  cost  of  fencing  the 
remaining  three  sides  may  be  the  least  possible  ? 

28.  Two  towns,  A  and  B,  are  on  opposite  sides  of  a  river  6 
miles  wide,  and  B  is  10  miles  below  A.  A  person  can  walk  along 
the  shore  twice  as  fast  as  he  can  row  across.  At  what  point  must 
he  leave  the  shore  so  as  to  get  from  A  to  B  in  the  shortest  time  ? 


9.    In  the  equation  —-\-—(l-  ^V=  i,  find  the  relation  be- 
- 2  ^        ^2  I 


a-      (f^ 


tween  ^,  g,  a,  and  h  when  the  quadratic  in  x  has  equal  roots. 

30.  What  are  limits  between  which  x^  —  5  ic  +  5i  is  negative  ? 

31.  If  a  and  /3  denote  the  roots  of  x'^  ■\- px -\- q  =  Q^  find  in 
terms  of  p  and  q  the  value  of  — 

i.   a  +  iS.  iii.   a2+i32.  V.  i  +  i- 

ii.   ajS.  iv.  -  +  -•  vi.   a^  +  0^, 

a      ^ 

32.  If  the  height  of  the  thermometer  is  expressed  by  the  func- 
tion x'^  —  2x  —  29 J,  where  x  denotes  the  number  of  days  counted 
from  a  fixed  time,  for  how  many  days  will  the  thermometer  be 
below  zero  ? 

126.   Every  equation  of  the  form 
can  be  solved  as  a  quadratic,  and  be  put  under  the  form 


ic**  =  i(— i?  ±  Vp^  -  4g). 


148  THE  QUADRATIC. 

For,  put  aj"  =  y,  and  the  equation  becomes 

whence  the  solution  follows. 
Ex.  1.  a;8-3x*- 208  =  0. 

Hence  a:*  =  |(3  ±  29)  =  16,  or  -  13. 

.-.  cc2  =  +  4,   -  4,  +  V13>  -  Vl^- 
x  =  +2,  -2,   +2i,  -2 1,  ±  Vi'Vl^;  ibV- V12; 
which  gives  the  8  roots. 

Ex.  2.  xi  -  7  a^t  -  8  =  0. 

Let  y  z=  x^,  then  X2  =:  2/^. 

.-.  y^  -7y  -S  =  0,  and  ?/  =  -  1,  or  8. 

/.  xi  =-1,  or  8,  and  x^  =  1,  or  4096. 

/.  X  =  ^1,  or  16. 

127.   Every  equation  of  the  form 
can  be  solved  as  a  quadratic,  and  exhibited  in  the  form 


fx  =  i{-p±Vp^  —  4:q). 

Whence,  if  /ic  =  0  is  solvable,  the  equation  can  be  com- 
pletely solved. 

Ex.  1.    Given  (x^  -  x  ~  1)2  +  4(a:2  _  x)  -  6  =  0. 
This  can  be  put  into  the  form  — 

(x2  -  X  -  1)2  +  4(x2  _  x  -  1)  -  2  =  0. 

...  x2  -  X  -  1  =  K- 4  ±  V24)  =  - 2  ±  V6. 
Then  x^  -  x  +  I  T  V^  =  0. 


Whence  x  =  i(l±V±  4 V6  -  3). 


IRRATIOKAL  EQUATIONS.  149 

On  account  of  the  double  square  root,  we  have  by  permutation 
of  signs  4  values  for  x,  in  all,  as  we  should  have,  since  x  rises  to  the 
4th  power  in  the  expanded  equation. 

Ex.  2.  x2-2x  +  V(^^-2a^  +  6)=6. 

Add  6  to  each  side ;  then 

x2  _  2  ic  +  6  +  y/{y?'  -  2  ic  +  6)  =  12. 

/.  V(^^  -  2  X  -f  6)  =  -  4,  or  +3. 

And  ic2  _  2  a:  +  6  =  16,  or  9. 

Whence  x  =  3,  -  1,  1  ±  V^- 

IRRATIONAL    EQUATIONS. 

128.  An  equation  which  involves  the  variable  under 
a  root  sign  is  called  an  irrational  equation. 

These  may  always  be  freed  from  irrationality,  and 
presented  as  rational  equations ;  but  the  rationalizing  of 
them  introduces  certain  uncertainties  of  solution  which 
it  seems  impossible  to  avoid. 

A  few  examples  will  make  this  clear. 

Ex.  1.    Given  y/a  +  -^x  =  ^ax. 

This  is  readily  reduced  by  dividing  throughout  by  y/x,  or  by 
treating  the  equation  as  having  ^/x  as  its  variable.    Then 

y/x{y/a-V)  =  y/a, 
and  ^^         a 


and  the  solution  is  exact. 

Ex.  2.    Given  Va  -\-  x  ■{■  Va  —  x  =  2y/x. 

As  it  is  always  profitable  to  reduce  the  number  of  terms  con- 
taining Vx,  where  possible,  we  may  divide  throughout  by  y/x^  and 
obtain  — 


ve^o-^ve-o=^- 


150  IRRATIONAL   EQUATIONS. 

Let  -  —  z^  and  square  ;  then 

X 

Divide  by  2,  transpose  z,  and  square,  and 

Z^-1=^-4:Z  +  Z^. 

/.  ^  =  CO  (Art.  77),  and  5?  =  J. 

Whence  x  =  0,  and  i^. 

6 

The  root  -— -  satisfies  the  irrational  equation ;  but  the 
o 

root  x  =  0  does  not  satisfy  the  equation,  and  although 
obtained  by  the  legitimate  transformations  of  algebra,  is 
not  really  a  root  of  the  given  equation,  as  the  test  of  a 
root  is  that  it  shall  render  the  equation  an  identity  when 
substituted  for  the  variable.  The  root  x  =  0,  however, 
satisfies  the  equation  Va  -{-x  —  Va  —  x  =  2-yJx,  which 
differs  from  the  other  in  a  single  sign. 

The  probable  explanation  of  this  peculiarity  is  that 
owing  to  the  disappearance  of  certain  negative  signs,  in 
rationalizing  the  equations  by  squaring,  both  these  equa- 
tions reduce  to  the  same  rational  form ;  and  as  far  as 
this  form  is  concerned,  there  is  nothing  by  which  we  can 
know  from  which  of  the  two  irrational  equations  it  has 
come. 

Hence  there  is  no  reason  why  the  roots  obtained  should 
not  satisfy  one  of  the  irrational  equations  as  well  as  the 
other.  But  it  is  quite  evident  that  both  roots  cannot 
satisfy  both  irrationals. 


Ex.  3.  V^  +  Va  -  ^ax  +  x^  =  ^a. 

Transposing  yjx^  squaring,  and  cancelling  a, 


y/ax  +  ic2  =  2  y/ax  - 


IRRATIONAL  QUADRATICS.  151 

-yjx  is  a  divisor  of  this.    Hence  (Art.  76)  Vx=0,  and  5c=0,  and 

Va  +  X  =  2^a  —  ^x. 
Squaring,  a  +  a;  =  4a  +  x-  4\/ai. 

.-.  X  =  Qo  (Art.  81,  Cor  2),  and  3a  =  4\/aa;. 
Whence  x  =  j^  a. 

Of  these  roots,  0  and  ^^  a  satisfy  the  given  irrational  equation, 
while  ic  =  00  satisfies  the  equation 

y/x  —  Va  +  \/ax  +  ic2  =,^a, 

in  which  the  signs  before  two  root- symbols  are  changed. 

Ex.  4.  Given  Zx-\r  VSOa;-  71  =  5. 

Transposing  3  x  and  squaring,  we  obtain 

ic  =  4,  or  2|. 

Neither  of  these  roots  satisfies  the  given  irrational  equation, 
they  being  roots  of 

3cc-V30x-71  =  5. 

Whether  the  given  equation  has  an  expressible  root  or  not,  it 
cannot  be  found  by  the  usual  methods  of  solution. 

EXERCISE  IX.  c. 
1.  Solve  the  following  — 


i.  a  +  «  +  V2ax-f  «2  =  &.        iii.   V4 a  +  sc  =  2 V6  +  x  -  V^i 


ii.  a-\-x-\-  Va2  -f  x^  =  h.  iv.   ^a-\-x  +  Va  —  x  =  2^0;. 


V.    \/aTx=2?«/{x2  +  6a&  +  62). 

2.  Find  X,  when  ^  "  ^^ .  ^/fli^'^'l  =  1. 

l  +  ax    AfVl-tiK/ 

3.  Find  X,  when  ^^"+^  =  1+VlT^. 

Vl-x     l-Vl-x 


152  IRRATIONAL  QUADRATICS. 

4.  Findic,  when  1+^(1 --)=J(l  + -V 


6.  Find  re,  when  a  -\-  x -}-  V2  ax  -\-  x^  =  b^{a  +  x  —  V2  ax  +  x^}. 

6.  Solve  the  equation  6 x  —  4V6x  +  1  =  x^  _  2  ^  —  4. 

7.  Solve  the  equation  x(-y/x  +  1)2  =  8(ic  +  ^x)  +  240. 

8.  Solve  the  equation  a  +  xVl  -\-  a^  =  aVl  —  x^  +  xVl  —  a"^. 

9.  Solve  the  equation  9 x  +  8  +  2  xV9x-\-4  =  15  ^2  +  4. 

10.  Solve  the  equation  x  +  V(»^^ -  ax+b^)=—+b, 

a 


CHAPTER   X. 

Indeterminate  and  Simultaneous  Equations 
OF  THE  First  Degree.  —  Simultaneous  Quad- 
ratics. 

129.  When  a  positive  integral  equation  contains  a 
single  vaiiable,  the  value  or  values  of  that  variable  may 
be  found,  theoretically  at  least,  in  terms  of  the  constants. 
But  if  the  function  contains  two  variables,  the  value  of 
either  will  contain  not  only  constants,  but  the  other 
variable,  and  thus  this  value  will  not  be  constant,  but 
variable,  and  therefore  arbitrary. 

Thus  if  3a; -1-22/ =  6,  x  =  2  —  ^y,  and  y  may  take  as 
many  values  as  we  please,  and  to  every  value  of  y  will 
correspond  a  single  value  of  x ;  and,  conversely,  to  every 
value  of  X  will  correspond  a  single  value  of  y.  Such 
equations  are  accordingly  called  Indeterminate,  and  as  we 
have  seen  in  Art.  117,  they  have  a  graph  which  is  a 
straight  line. 

The  study  of  Indeterminate  equations  is  practically  the 
study  of  their  graphs,  and  as  a  consequence  Indeter- 
minate equations  in  relation  to  their  graphs  form  the 
subject-matter  of  the  great  body  of  higher  geometry 
known  as  Analytic  Geometry. 

130.  Linear  Indeterminate  equations  are  considered 
here  only  under  the  restrictions  that  the  corresponding 
values  of  x  and  y  shall  be  positive  integers. 

153 


154  IKDETERMINATE   EQUATIONS. 

The  subject  is  best  discussed  in  examples. 

Ex.  1.  To  find  positive  integral  values  for  x  and  y  which  shall 
satisfy  the  equation  7  ic  +  11  ?/  =  103. 

This  gives  x  =        ~ ^,  which  is  to  be  integral. 

But  ^Q^  -^^y  =  14  _  y  +  ^-^y .  and  as  y  is  to  be  integral, 
14  —  2/  is  integral,  and  therefore  —^ — ^  is  integral. 

Also,   as  the  product  of  integers  is  integral,   2  x  —^ — ^,  or 

1--^^ — ^  is  integral ;  i.e.     ~  ^  is  integral. 

The  purpose  in  multiplying  by  2  is  to  make  the  coefficient  of  y 
greater  by  1  than  a  multiple  of  7,  so  that  after  casting  out  all  inte- 
gers the  coefficient  of  y  may  be  1. 

Now,  put  -  ~  y  =zp.  Then  y  z=S  +  7p;  and  putting  this  value 
for  y  in  the  original  equation  gives  x  =  10  —  11^. 

Therefore  a;  =  10  — 11^^ 

V  IS  the  general  solution. 
y  =  S  +  7p     / 

The  particular  solutions  are  got  by  giving  to  p  any  allowable 
Integral  values,  provided  such  values  do  not  make  x  or  y  negative. 

We  readily  see  that  in  the  present  question  p  can  have  only  one 
value,  zero  ;  and  x  =  10,  y  =  3,  is  the  only  solution. 

Ex.  2.  Can  1 1  be  paid  in  9-cent  pieces  and  7-cent  pieces,  and  if 
so,  how  ? 

Let  X  =  the  number  of  9-cent  pieces,  and  y  =  the  number  of 
7-cent  pieces. 

Then  9x-\-7  y  =  100  is  our  equation. 

The  solution  gives  y  =  4:  —  9p  and  x  =  S  -\-7p. 

When  p  =  0,  -1, 

X  =  8,       1, 
2/ =  4,     13. 

There  are  thus  two  solutions,  one  by  8  9-cent  pieces  with  4  7-cent 
pieces ;  the  other,  by  1  9-cent  piece  with  13  7-cent  pieces. 


INDETERMINATE   EQUATIONS.  155 

131.  It  will  be  noticed  that  in  the  general  solution 
the  coefficient  of  p  in  the  value  of  x  is  the  coefficient  of 
y  in  the  equation,  and  the  coefficient  of  p  in  the  value  of 
y  is  that  of  x  in  the  equation,  one  of  the  signs  being 
changed. 

Hence  in  the  equation  ax  —  by  =  c,  p  will  have  the 
same  sign  in  the  values  of  both  x  and  y,  and  the  number 
of  solutions  will  be  unlimited. 

Ex.  To  find  solutions  of  7  x  —  5  y  =  23,  we  easily  obtain 

X  =  4  +  5j9,  2/  =  1  -f  7 p. 
Therefore  when  ^  =  0,  1,    2,     3,     4... 

ic  =  4,  9,  14,  19,  24... 

2/  =  l,  8,  15,  22,  29... 

132.  In  the  equation  ax  ±by  =  c  there  can  be  no 
solution  in  positive  integers  if  a  and  b  have  a  common 
factor  which  is  not  also  a  factor  of  c. 

Eor,  let  a  =  mf,  and  b  =  nf. 

Then  ax  ±by  =  mfx  ±  nfy  =  c ; 

and  mx  ±ny  =  — 

But  -  is  a  fraction  by  hypothesis,  and  is  not  the  sum 
or  difference  of  two  integers. 

133.  The  following  problem  is  nearly  related  to  one 
of  the  three  preceding  articles. 

Ex.  To  find  an  integral  number  which  when  divided  by  3  leaves 
1,  by  5  leaves  4,  and  by  7  leaves  2. 

If  X  denotes  the  number,  x  is  evidently  of  any  one  of  the  forms 


156  INDETERMINATE   EQUATIONS. 

3m  +  1,  5n  +  4,  or  7p  +  2,  and  these  are  to  represent  the  same 
number ;  we  have 

3m  +  l  =  5n  +  4=:7i)  +  2. 

/.  m  =    ^  "^  '  =  an  inte^?. ,  and  -  =  an  integ.  =  m, 
3  °  3  b  > 

and  n  =  Sm. 

7  w  —  2 

Again,  15 m  +  4  =  7p  +  2,  or  w  ==  -^ =  an  integ. 

15 

»  —  11 
■    ^  =  an  integ.  =  m,  and  ^  =  15  m  +  H. 


15 

Hence  x=7p  +  2  =  105  m  -t-  79,  which  is  the  general  solution. 
If  m  =  0,  we  have  79  as  the  lowest  number  satisfying  the  conditions. 

The  following  method  of  solution  is  also  convenient.  Let  x  be 
the  number. 

Then  ^^^=^,   ^-^,  and  ^^=^  are  all  integers.     Put  ^^  =  », 

3  5  7  "^  3         ^' 

and  X  —  375  +  1.     Substitute  this  value  of  x  in  the  second  fraction, 

and     ^  ~  '    is  an  integer,  or  ^  ~^     is  an  integ.  =  q. 
5  5 

,'.  p  =  6q  -{-  1^  and  x  =  15 g  +  4. 
Substitute  this  new  value  of  x  in  the  third  fraction,  and 
— ^  "*"      is  an  integ.,  or  ^^ —  is  an  integ.  =  r. 

Then  g  =  7  r  —  2,  and  x  =  105  r  —  26,  which  is  the  general  solu- 
tion,    r  =  1  gives  79  as  the  lowest  number  satisfying  the  conditions. 


EXERCISE  X.  a. 

1.  Find  positive  integral  solutions  to  the  following  — 

i.    Sx+7  y  =  10l.  iii.  455c  -  13?/ =  38. 

ii.    lSx+ny  =  200.  iv.   x-ll?/  =  48. 

2.  Find  multiples  of  23  and  15  which  differ  by  1 ;  which  differ 
by  2;  by  3. 

3.  How  can  I  measure  off  a  length  of  4  feet  by  means  of  two 
measures,  one  7  inches  long  and  the  otlier  13  inches  long  ? 


LINEAR    SIMULTANEOUS   EQUATIONS.  157 

■  4.   I  have  nothing  but  4-pound  and  7-pound  weights.     How  can 
I  weigh  exactly  45  pounds  ? 

5.  A  company  of  soldiers  when  arranged  4  abreast  lacks  1  man, 
when  5  abreast  it  lacks  2  men,  when  6  abreast  it  has  3  too  many, 
and  when  7  abreast  it  forms  a  complete  block,  llow  many  men,  at 
least,  are  in  the  company  ? 

6.  A  wall  27  ft.  9  in.  long  is  to  be  panelled  with  two  widths  of 
boards,  8  in,  and  5  in.  wide.  How  many  of  each  kind  must  be 
used  so  that  — 

i.  The  narrow  boards  may  exceed  the  wide  by  the  least  number 
possible  ? 

li.  The  wide  may  exceed  the  narrow  by  the  least  number 
possible  ? 

lii.   The  whole  number  of  boards  may  be  the  least  ? 


LINEAR    SIMULTANEOUS    EQUATIONS. 

134.  The  equations  ax  -\-hy-^c=0  and  aiX-{-biy-{-Ci=0 
are  both  indeterminate,  and  being  linear,  both  have 
straight  lines  as  their  graphs  (Art.  117,  Ex.  1). 

These  straight  lines  have  some  common  point,  their 
point  of  intersection,  and  at  this  point  the  corresponding 
values  of  x  and  y  must  be  such  as  to  satisfy  both  equa- 
tions ;  and  as  the  graphs  have  only  one  common  point, 
there  is  only  one  such  set  of  values. 

When  two  equations  are  given,  and  it  is  required  to 
find  corresponding  values  of  x  and  y  that  shall  satisfy 
both,  the  equations  are  called  simultaneous,  and  these 
particular  values  of  a;  and  y  form  the  solution  to  the  set 
of  two  equations. 

Ex.  The  two  equations  2x-\-Sy  =  9  and  Sx-h  2y  =  11  are 
satisfied  by  the  values  x  =  3,  y  =  1,  and  by  no  other  values. 


158  LINEAR   SIMULTANEOUS   EQUATIONS. 

135.  Problem.  To  solve  a  set  of  two  simultaneous 
equations  with  two  variables.  The  methods  will  be  ex- 
plained through  examples. 

Let  4:X  —  3y  =  26  and  3x  -\-  5y  =  5  be  the  equations. 

First  method.  — By  addition  and  subtraction. 

Add  5  times  the  first  equation  to  3  times  the  second, 
and  the  coefficients  of  y,  being  equg]  with  opposite  signs, 
disappear,  and  we  have  left 

29  X  =  145 ;  whence  x  =  B. 

Again,  to  get  rid  of  x,  we  subtract  4  times  the  second 
equation  from  3  times  the  first,  and  obtain 

—  29  ?/  =  58 ;  and  hence  y  =  —  2. 
Second  method,  —  By  substitution. 
The  first  equation  gives  0?  =  — — — ^;  and  substituting 
this  for  X  in  the  second  gives 

I§-±-^  +  5y  =  5,  or  78  +  29y  =  20. 
4 

Whence  29  y  =  -  58,  and  ?/  =  -  2. 

Next  substitute  —  2  for  2/  in  either  of  the  equations, 

and  we  get  the  value  of  x. 

Third  method.  —  By  comparison. 

The  values  of  x  found  from  the   two  equations  are 

2Q4-3v  5 5?/ 

— — — ^  and  ^;  and  as  these  must  be  equal,  we  have 

4  3      '  ^ 

78  +  92/  =  20  -  20y,  or  2/  =  -  2. 

Similarly,  y  = —  ==  — 

/.  20a;  —  130  =  15  —  Oaj,  or  a?  =  5. 


LINEAR   SIMULTANEOUS   EQUATIONS.  159 

Fourth  method.  —  By  an  arbitrary  multiplier. 
Multiply  one  of  the  equations,  the  first,  by  an  arbitrary 
multiplier  m,  and  add  to  the  other. 

We  have     (4  m  +  3)  a;  — (3  m  —  5)  2/  =  26  m  +  5. 
As  m  is  arbitrary,  we  may  give  to  it  such  a  value  as 
will  make  either  of  the  brackets  zero. 

If  m  =  f ,  (-2/  +  3)  a:  =  26  X  1 4-  5,  or  aj  =  5. 
If  m  =  —  -|,  we  similarly  obtain  ?/  =  —  2. 

EXERCISE  X.  b. 

1.  7(a;  _  5)  =  2^  _  2,  4  ?/  -  3  =  i(x  +  10),  to  find  x  and  y. 

2.  (x+5)(2/+  7)=:(aj-l)(2/-9)+104,  and2a:  +  10  =  32/  +  l, 
to  find  X  and  y. 

3.  X  —  a  =  c(y  —  b) ,  a(x  —  a)  +  b(y  —  b)+  abc  =  0,  to  find  x 
and  y. 

4.  ^  +  ^  =  l=:^Z^  +  ^^,tofinda:and2/. 
a     b  b  a 

5.  2  s  =  n(«  +  z),  d{n  —  1)  =  2;  —  a,  to  find  a  relation  not  con- 
taining n. 

6.  A  and  B  have  $500  between  them.  A  gains  from  B  1  of  B's 
money  and  |50.  B  then  gains  from  A  J  of  A' s  original  money 
and  $'50,  and  they  then  have  the  same  amount.  What  had  they 
to  begin  with  ? 

7.  A  fraction  is  such  that  if  2  be  added  to  its  numerator  it 
becomes  i,  and  if  1  be  added  to  its  denominator  it  becomes  |. 
Find  the  fraction. 

8.  A  number  of  two  digits  has  the  sum  of  the  digits  12,  and  if 
6  times  the  first  digit  be  subtracted  from  the  number,  the  digits  are 
exchanged.     Find  the  number. 

9.  There  are  two  kinds  of  coin  such  that  a  and  b  pieces  respec- 
tively are  equal  to  |1.  How  many  pieces  of  each  kind  must  be 
taken  so  that  the  value  of  c  pieces  together  may  be  one  dollar  ? 


160  SETS    OF   LINEAR   EQUATIONS. 

10.  A  farm  was  taxed  at  30  cents  an  acre,  and  the  tenant  being 
allowed  10  %  off  his  rent  found  the  allowance  to  amount  to  15  dollars 
more  than  the  taxes.  The  next  year  the  taxes  were  doubled,  and 
the  farmer  was  allowed  15  %  off  his  rent,  which  just  paid  his  taxes. 
What  was  the  rent  of  the  farm,  and  how  many  acres  did  it 
contain  ? 

SET    OF    THREE    LINEAR    EQUATIONS. 

136.   Denote  the  three  equations  by  A,  B,  and  C,  and 

let  the  variables  be  x,  y,  z. 

Find  from  A  the  value  of  x  in  terms  of  y,  z,  and  the 
constants,  and  substitute  this  value  for  x  in  equations 
^  and  C.  We  are  then  said  to  have  eliminated  x  between 
A  and  B,  and  also  between  A  and  (7;  and  we  have  two 
new  equations,  D  and  Ej  which  contain  only  y  and  z  as 
variables. 

Thus  in  eliminating  one  variable  we  reduce  the  num- 
ber of  our  equations  by  one. 

Now  eliminate  y  between  D  and  E,  and  we  are  left 
with  a  single  equation  F,  which  is  just  sufficient  to 
determine  z  in  terms  of  the  constants.  Hence  we  readily 
find  2/  and  x. 

Ex.   Let  ^be      x  +  Sy  —  2z  =  l, 

Bhe    Sx  —  2y  -\-     ^  =  5, 
Cbe    2x  +  4y-Sz  =  l. 

From  A  we  have  x  =  l  —  Sy-\-2z. 

Substituting  this  value  for  x  gives  — 

inB,  S-9y  +  6z-2y-\-     z  =  5,  or-Uy+7z  =  2  (D) 

inC,  2-6i/  +  4^  +  4?/-3^  =  l,  or-    2?/+     z  =  -l      (E) 

And  eliminating  y  between  D  and  E  gives 
3  ^  =  15,  or  ^  =  5. 


SETS    OF   LINEAR    EQUATIONS.  161 

Thence  E  gives  y  -  ^-^  =  3,  and  x  =  l-94-10  =  2. 

,'.  X  =  2^  y  =  Sj  z  =  6  is  the  solution. 

This  method  of  eliminating  x  and  y  is  always  sufficient, 
but  it  may  not  always  be  the  most  convenient,  and  any 
method  of  elimination  will  answer  if  carried  out  in  proper 
form.  Thus  whatever  process  of  elimination  be  employed, 
the  same  letter  must  be  eliminated  between  one  of  the 
equations  and  each  of  the  others.  The  following  method 
is  very  convenient  and  gives  the  same  results  as  the  last. 

Subtract  jB  from  3  A  This  gives  11  ?/ -  7^  =- 2  ...(D) 
Subtract  C  from  2  A.  This  gives  2y  -  z  =  l  .  .  .  .  (E) 
Subtract  11  {E)  from  2  (Z>),  and  -  3  2;  =  -  15,  or  z  =  ^. 

137.  From  the  preceding  article  it  appears  that  when 
we  eliminate  a  variable  from  a  system  of  equations  we 
lose  one  equation ;  and  conversely,  that  by  combining 
one  of  the  equations  with  each  of  the  others  of  the  set 
we  can  eliminate  one  variable  from  the  set. 

Hence  the  two  following  results : 

(1)  That  n  equations  are  just  sufficient  to  determine 
n  variables ;  and  conversely,  that  n  variables  can  be 
determined  from  a  system  of  n  equations,  provided  the 
equations  be  independent,  i.e.  such  that  any  one  cannot  be 
derived  from  the  others. 

(2)  That  with  n  variables,  and  n  —  1  equations,  the 
final  result  will  be  a  single  equation  containing  two 
variables,  and  be  thus  indeterminate.  And  hence  a  system 
of  71  —  1  equations  with  n  variables  is  indeterminate. 

Thus  the  two  equations  with  three  variables, 

3x-}-2y  —  6z  =  4.,   and  4:X  +  y-^z  =  10, 


162  SETS   OF   LINEAR   EQUATIONS. 

give,  by  eliminating  x,  the  single  indeterminate  equation 

5y  -  27 z  =  36. 

If  this  be  solved  for  positive  integers,  we  may  find 
one  or  more  systems  of  positive  integral  values  for  Xj  y, 
and  z,  which  will  satisfy  the  two  given  equations. 

138.  Let  there  be  n  variables  and  n  + 1  equations ; 
an  important  case  requiring  consideration. 

As  the  n  + 1  equations  are  sufficient  to  determine 
n  +  1  variables,  we  make  up  this  number  of  variables  by 
taking  one  of  the  constants,  provided  it  be  literal,  and 
considering  it  as  a  variable.  After  eliminating  the  n 
true  variables,  we  have  a  single  equation  in  which  this 
pseudo-variable  is  the  only  one  occurring;  or,  in  other 
words,  we  have  a  single  equation  expressing  a  necessary 
relation  amongst  the  constants.  This  equation  is  called 
the  Eliminant  of  the  system. 

Ex.  Given  3ic  +  2?/  =  a,  2x  —  4y  =  &,  and  x  +  5 ?/  =  c,  to  find 
the  eliminant. 

Eliminating  x  gives 

13  2/  =  3  c  —  a,  and  14  ?/  =  2  c  —  5. 

And  now  eliminating  ?/,  we  have 

14  a  -  13  6  -  16  c  =  0, 

as  the  necessary  relation  between  «,  5,  and  c  that  the  three  equa- 
tions may  be  compatible. 

If  a,  h,  and  c  are  numbers  which  do  not  satisfy  the 
eliminant,  the  equations  are  incompatible,  and  cannot  be 
satisfied  by  any  one  set  of  corresponding  values  of  x  and  y. 

On  the  other  hand,  if  a,  ft,  and  c  are  numbers  which 
do  satisfy  the  eliminant,  one  of  the  equations  is  derivable 


SETS   OF   LINEAR   EQUATIONS.  163 

from  the  other  two,  and  thus  expresses  no  relation  but 
what  is  already  given  by  the  other  two.  The  equations 
are  then  not  independent,  and  one  of  them  is  redundant 

139.  An  important  system  is  one  of  homogeneous 
equations,  in  which  the  number  of  variables  is  greater  by 
one  than  the  number  of  equations. 

Let  aiX-\-  h^y  +  Cjs;  =  0  =  a^x  +  &22/  +  ^^  ^®  such  a  sys- 
tem. 

Dividing  through  by  y,  these  take  the  form 

«i- +  Ci-  +  &i  =  0  =  a2-  +  C2  -  +  h, 

y       y  y       y 

which  is  a  system  of  two  equations,  with  the  two  variables 
X :  y  and  z :  y. 


Solving,  we  get  -  =  h^i^Z^. 

y     Cittg  —  c^ai 

a;         _  ?/         _  ; 


-,  by  symmetry. 


61C2  —  h2Ci      Cia2  —  c^ai      a^^g  —  ajbi 

Or         x:y'.z  =  biC^  —  ^2^1  •  Cia2  —  C2ai :  aib2  —  ajbi. 

And  the  variables  are  any  quantities  proportional  to 
the  denominators  of  the  fractions. 

Ex.   Let  2x  +  6?/-3;s=:0  =  4x-32/  +  2r. 

Then  x :  ?/ :  ^  =  3  :  14  :  30, 

and  the  numbers  3,  14,  and  30,  or  any  multiples  of  these,  will 
satisfy  the  equations. 

If  in  these  equations  we  make  2;  =  a  constant,  both  x 
and  y  take  fixed  values,  and  we  have  the  common  case 
of  two  equations  with  two  variables. 


164  SETS   OF  LINEAR   EQUATIONS. 

140.     Problem.     To  solve   a   system   of  three   lineal 
equations  with  three  variables  by  arbitrary  multipliers. 
Let  the  system  be 

biX  +  6jj2/  +  h^  =  ^2? 
CiX  +  C22/  +  c^z  =  ds, 

Multiply  the  equations  by  the  arbitrary  multipliers 
I,  m,  and  n,  respectively,  and  add;  then 

(lai-\-mhi-\-nc^x-\-  (la2-{-mh2-\-nc^y-\-  {la^-\-mh^-\-nc^z 

=  Id^  +  mdg  +  ^^^3- 

As  I,  m,  n  are  arbitrary,  we  may  so  take  their  values 
as  to  make  any  two  of  the  brackets  zero. 
Thus  to  eliminate  y  and  z  we  must  have 

The  solution  of  this  is,  by  the  preceding  article, 
I         _        m        _         n 
^2^3  —  ^3^2      ^2^X3  —  (^^(^2      ^2^3  —  ^'3^2 

And  Z,  m,  ti  are  any  quantities  proportional  to  the 
denominators.  Naturally  we  take  as  the  multipliers  the 
denominators  themselves. 

The  reader  will  find,  upon  trial,  that  these  multipliers 
cause  the  coefficients  of  y  and  z  each  to  become  zero. 

We  notice  that  the  multiplier  for  any  equation  does 
not  contain  any  coefficient  from  that  equation  or  any 
coefficient  of  the  variable  to  be  determined.  Thus  the 
multiplier  for  the  first  equation  is  ^2^3  —  &3C2,  and  does 
not  contain  a  suffix  1,  or  an  a,  etc.  A  little  observation 
on  the  forms  of  these  multipliers  is  better  than  any 
description. 


SETS   OF   LINEAR   EQUATIONS.  165 

Ex.  1.   Given  x  +  2ij~j-Sz  =  9, 

2x+     y+     z  =  U, 
Sx  +  2y  +  ^z  =  S. 
To  eliminate  y  and  z  the  multipliers  are 

Z  =  3,  m=  — 4,  n=— 1. 
.\  Sx-Sx-Sx  =  27  -56-S, 
or  —  8  X  =  —  32,  and  x  =  4:. 

To  eliminate  x  and  y  the  multipliers  are 
1  =  1,  m  =  4,  n  =—  S. 
.'.  3^  +  42; -16^  =  9  + 56 -9, 
or  —  8  ^  =  56,  and  z  =  —7, 

Ex.  2.  Given  ax  +  ?/  +  ^  =  0, 

ic  +  a?/  +  2J  =  1, 
«  +  2/  +  «^  =  -  1. 
The  multipliers  for  eliminating  y  and  ^  are  a2_i^  1  _^^  and  1  —  a. 
Thence,  {a(a^  -  1)  +  2(1  -  a)}x  =  0,  and  x  =  0. 

Similarly,  we  find  y  = ,  z  = 

a  —  1  1  —  a 

141.  With  four  or  more  equations,  multipliers  may 
also  be  found  which  will  eliminate  all  the  variables  but 
one,  but  these  multipliers  are  too  complex  for  convenient 
use.  In  the  chapter  on  Determinants  it  is  shown  that 
all  sets  of  linear  equations  are  solvable  upon  the  same 
general  principle. 

A  set  of  four  equations  may  be  dealt  with  as  follows : 

Ex.   Let  them  be         x  +  y+z-\-u=4: (A) 

x  +  2y-\-Sz  +  ^u  =  10 (B) 

2x-\-    y  +  Sz+    u=    1 (C) 

3x  +  3?/  +    ^  +  2  w  =  10 (D) 


166  SETS   OF   LINEAR   EQUATIONS. 

Take  the  first  three  equations,  and  consider  the  x  and  y  part  as 
forming  a  single  term. 

The  multipliers  for  eliminating  z  and  u  are  —  9,  2,  1  ;  and  these 
give  5  X  +  4  2/  =  9. 

Now  take  the  last  three  equations ;  the  multipliers  are  5,  —  2, 
—  9  ;  and  these  give  26  x  +  19  ?/  =  54. 

From  these  two  new  equations  we  find  cc  =  5,  ?/  =  —  4.  The 
values  of  u  and  z  are  then  readily  found  to  be  z  =—  \^  w  =  4. 

EXERCISE  X.  c. 

1.  Solve  the    set,    2x  +  4  ?/ +  5^  =  49,    3x  +  5?/ +  6^;  =  64, 
4x  +  32/  +  45;  =  55. 

2.  Solve  the  set,   2x  —  Sy  +  z  =  2,  x  +  y  —  2z  =  1^  Sx  +  2y 
^Sz  =  6. 

3.  Solve  the  set,  x  —  y  —  2z  =  S,  2x  +  y  —  Sz  =  l\f  Sx  —  2y 

+  z  =  i. 

4.  Solve  the  set,  ax  +  by  —  az  =  b{a-\-b),  bx  —  ay-\-z=b(b  —  a), 
x-\-2y  —  2z  =  4:X  —  b. 

5.  Solve  the  set,  x  +  2/ +  ^  =  0,  (a  +  b)x  +  (b  +  c)y+(c-i-a)z=Of 
abx  +  bey  +  caz  =  1. 

6.  Solvetheset,  ?-A  +  l  =  7f,  A.  +  J_  +  2^ioj,   A__L 
^  x     by     z  3x     2y      z  ox     2y 

7.  If  2x  -\-  Sy  =:  a,  x  —  y  =  by  x  +  2y  =  c,  find  the  eliminant. 

8.  Find  the  eliminant  of  ax-\-y  =  1,  bx  +  S  y  =  6,  ex  -\-  b  y  =  10. 

9.  Find  the  eliminant  of3x  +  2?/  +  a=0,    x  —  Sy  +  b  =  Oj 
2x  +  y  —  c  =  0. 

10.  Solve    the    set,    Sx-2y  +  6z  =  U,    2x  +  y  -Sz  =  \0, 

Sx  —  Sy-]-2z  =  SS;  and  explain  the  cause  of  any  difficulties. 

11.  If  ttiX  +  b^y  +  c^z  =  a.^x  +  6.^?/  +  c.^z  =  a.^x  -^  b.^y  +  c^z  =  0, 
show  that  aiC^^Cg  —  b.^c.^)  +  &i(c2«3  —  c^a.^)  +  <^i{P"p6  —  (^M  =  ^' 


SIMULTANEOUS  QUADRATICS.  167 

12.  When  x -{- y  —  z  —  u  =  2x—2y -\- z-\-u=Sx —  y -{•Sz—u=0, 
find  four  numbers  having  the  ratios  xiy.ziu. 

13.  What  does  Ex.  12  reduce  to  when  m  =  1  ? 

14.  Solve  the  set,  a +  26  +  3c+4(?  =  29,  4a  +  6+2c+3d:=23, 
3a  +  45  +  c  +  2(Z  =  25,  2a  +  36  +  4c  +  (?  =  23. 

15.  Solve  the  set,  x+  by  —  az  =  -<,  ax  +  y  — z  =  a%  -a;  -f  ay 
— « =  1  ^  o  a 

16.  Solve  the  set,  a'^x  +  ay  +  z  =  —  a^,  b^x  +  by  -^  z  =  —  b^^ 
c^x  +  cy  -}-  z  =—  c^;  and  reduce  the  values  of  the  variables  to 
lowest  terms. 

17.  Find  the  eliminant  of  ax  +  by  +  cz  =  bx  +  cy  +  az  =  ex 
+  ay  +  bz  =  0, 

18.  Solve  theset,  If^+lVl/^^.  IN      1/2  _  6  N      1/3^  IX 

x\z    yz)    y\x     zj     y\x     xzj     z\y     xj 

z\y     xyj     x\z     yj 

19.  Given    x(y  +  z)=  a^,    y(z-i-x)=  b\    z(x  +  y)  =  c^,    and 

—  +  --  =  —,  to  find  the  relation  connecting  a,  6,  c. 
a;2     2/2     ^'^ 

(Find  the  values  of  «,  t/,  and  z  from  the  first  three  equations, 
and  substitute  these  in  the  fourth.) 


SIMULTANEOUS    QUADRATICS. 

142.  A  system  consisting  of  one  quadratic  and  one 
linear  can  always  be  solved. 

The  most  general  type  of  a  quadratic  equation  of  two 
variables  may  be  written 

aoc^  +  %^  +  ^^  +  gx  +fy  +  c  =  0. 
And  any  linear  of  two  variables  may  be  written 


168  SIMULTANEOUS    QUADRATICS. 

If  we  substitute  for  x  from  the  linear  into  the  quadratic, 
we  have 

a  {py  +  ry  +  by^  —  hy  (py  -^r)-g{py  +  r)  -{-fy  +  c  =  0, 

a  quadratic  from  which  to  determine  y. 

Ex.  To  find  two  numbers  such  that  the  sum  of  their  squares 
and  their  product  is  a  and  the  sum  of  the  numbers  is  6. 

We  have,  x^  +  y"^  +  xy  =  a,  and  x  +  y  =  b^  where  x  and  y  de- 
note the  numbers. 

Substituting  for  x  from  the  hnear  into  the  quadratic, 

(p  _  yy  _|_  2/2  +  (^  _  2/)?/  =  a, 
or  y^  —  by  =  a  —  b'^. 

Whence  y  =  ^(b  ±  V4a-3  6-^), 

and  x  =  K^=FV4a-362). 

If  4  a  <  3  b^j  the  numbers  are  complex. 

The  equation  a^  -\- y^  -{- xy  =  a  and  x-\-y  =  b  are  sym- 
metrical in  x  and  y ;  and  whenever  this  is  the  case,  the 
values  of  x  and  y  must  be  interchangeable,  so  that  having 
the  two  values  of  y,  we  have  also  the  two  values  of  x. 

Thus  if  a  =  19,  and  &  =  5,  we  have  y  =  3  or  2,  and 
a;  =  2  or  3. 

143.  A  system  of  two  quadratics  with  two  variables 
does  not  in  general  admit  of  being  solved  as  a  quadratic, 
since  substituting  the  value  of  a  variable  from  one  of 
the  equations  into  the  other  will  in  general  give  rise  to 
an  equation  of  four  dimensions. 

Thus  the  system  x^  -{-y  =  a,  and  y^  -{-  x=b  gives,  by 
substituting  for  y^  x^ —  2ax^  -{- x  =  b  —  a^,  a  quartic  equa- 
tion. 

There  are,  however,  many  cases  in  which  a  sufficient 
relation  exists  between  the  forms  of  the  equations,  to 


SIMULTANEOUS   QUADRATICS.  169 

make  a  solution  possible  without  going  beyond  the 
quadratic. 

No  general  list  of  such  can  be  given,  and  no  very 
general  rules  of  procedure  can  be  laid  down  for  such 
cases  when  they  occur.  Practice  and  observation  are 
the  only  keys  to  success. 

The  following  are  given  by  way  of  illustration  : 

144.  When  two  quadratics  have  a  common  linear  fac- 
tor in  the  portions  involving  the  variables,  they  can  be 
solved. 

For  let  A  be  the  common  linear  factor,  and  let  C  and 
C  be  the  independent  terms. 

Then  the  equations  are  of  the  forms  AB  =  C  and 
AB'  =  C,  and  B  and  B'  must  be  linear  factors. 

Dividing  the  first  equation  by  the  second,  we  have 

B'    c'  a 

c 

And  as  —  is  a  constant,  one  variable  is  linearly  ex- 
pressible in  terms  of  the  other.  And  hence  by  substitu- 
tion we  obtain  a  quadratic  for  finding  one  of  the  variables. 


Ex.   Given 

Sx^ 

-4  2/2+    4 

[xy=z 

-21, 

12  0^2 

+  2?/2-l] 

xy  = 

-3. 

The  first  equation 

is     (Sx- 

-  2y){x  +  : 

^y)  = 

-21, 

and  the  second  is 

(Sx- 

-  2  ?/)  (4  X  - 

-y)  = 

-3. 

Dividing, 

x  +  2y 
^x-y 

=  7,  and  .• 

.  y  = 

3  a;. 

Substituting  in  the  first  equation, 

21  ^2  =  21 ;  and  x  =  ±  1. 
Thence  2/  =  ±  3. 


170  SIMULTANEOUS   QUADRATICS. 

145.  When  the  equations  are  homogeneous  in  the 
parts  involving  the  variables,  they  can  often  be  readily 
solved  by  putting  y  =  ux,  and  then  dividing  one  equation 
by  the  other.  This  gives  a  quadratic  for  finding  u,  and 
hence  a  known  linear  relation  between  the  variables. 

Since  u  will,  in  general,  have  two  values,  we  will  get 
two  quadratics  to  determine  x,  and  hence  x  will  have  in 
all  four  values,  as  it  should  have.  So  also  y  will  have 
four  values. 

Ex.   Given  sc^  +  xy  +  4  2/2  =  6, 

3  x2  _|_  8  2/2  =  14. 

Let  y  =  ux^  and  divide  equation  by  equation  ;  then 

3  x2  +  8  u^x^         7* 
or  4  ^2  +  7  ^  —  2,  and  u  =  —2,  or  J. 

.*.  y  =  —  2x,  or  \x;  and  substituting  these  values  in  one  of  the 
equations,  the  second  by  preference,  as  being  the  simpler, 

35c2  +  32a:2  =  14,  and  3x2  +  | x2  =  14. 

.-.  x  =  ±  WIO,   =±2, 
and  y  =  Tl^lO,  =Th 

146.  When  two  variables  are  involved  symmetrically, 
it  frequently  simplifies  the  solution  to  assume  two  new 
variables  whose  sum  shall  be  one  of  the  original  variables, 
and  their  difference  the  other. 

Ex.   Given  x^  -}-  y'^  -\-  x  +  y  =  S, 

x-^y  +  xy  =  6. 

The  variables  being  symmetrically  involved,  assume  a;  =  m  -f  t, 
and  y  =  u  —  V, 

Then  ^^2  ^  ^2  _j_     ^  _  4^ 

and  w2  _  ^2  ^  2  w  =  5. 


SIMULTANEOUS   QUADRATICS.  171 

Adding,  2u'^  +  Su  =  9;  and  u  =  f,  or  -  3. 

Substituting  these  values  for  u  in  one  of  the  new  equations, 
the  first,  we  get 


And  ic  =  2orl,   =-3db  i^2. 

And  y  being  symmetrical  with  x  has  the  same  values  ;  then 

x  =  2,  y  =  l;   x  =  l,  y  =  2;   x  =  -S-{-  i^2  ; 

2/=-3-V2;  x  =  -Z-i^2)   ?/=-3+iV2; 
are  the  four  sets  of  corresponding  values  of  x  and  y. 

The  present  equations  may  be   otherwise  solved  as 
follows : 

Add  twice  the  second  equation  to  the  first,  and  it  becomes 
(x  +  y)2  +  3(a:  +  2/)=18, 
whence  x  +  ?/  =  3,  or  —  6. 

Then  from  the  second,        xy  =  2,  or  11, 
and  (x  —  yy  =(x  +  yy  —  4:xy  =  1,  or  —  8. 

...  x-y  =  ±l,  or  2iV2. 
Whence        x  =  J(x  +  y  +  x  —  y)=2j  or  — 3  +  i^^  ; 


y  =  i(x-\-y-x-y)=l,  or  -S-i^2; 

and  the  values  of  x  and  y  being  interchangeable  give  the  four  val- 
ues as  before. 

147.  Various  devices  are  employed  to  obtain  solutions 
of  simultaneous  quadratics  and  other  simultaneous 
equations.  These  cannot  be  given  in  detail,  but  will  be 
illustrated  in  the  following  examples. 

Ex.  1.   Given  x^  -h  V^  =  275,  x-\-  y  =  6. 

(x  +  ?/)5  =  x5  +  5  xy  (2  x'^y  +  2  xy'^  -{■  x^  -\- y^)  +  y^  =  3125. 


172  SIMULTANEOUS    QUADRATICS. 

Subtracting  x^  -\-  y^  =  276  leaves 

5  xy  (x^  +  2x^y  +  2  xy'^  +  y^)  =  2850. 

.-.  x3  +  2  x:^y  -\-2xy^  +  y^  =  — • 
xy 

But  (x  +  ?/)3  =  a:3  +  3x2y  +  3ic2?/2  +  2/^  =  125. 

.-.  xy(x  +  ij)=6xy  =  126-  — • 
xy 

Whence  (xyj^  —  26xy  =  —114, 

and  xy  =  19,  or  6. 

Then  having  x?/  and  x  +  ?/,  we  readily  find 

a:  =  2,  or  3,  or  J  (5  ±  V^l); 
y  =  S,  or  2,  or  J  (5  T  iV^l)- 

Q  . 

Ex.  2.   Given     x  +  vx^  —  y'^  =  -  (Vx  +  ?/  +  Vx  —  ?/), 

2/ 
and  (x  +  ?/)  2  _  (x  -  ?/)  -^  =  26. 

Put  X  +  ?/  =  2  s2,  and  x  -  ?/  =  2  ^2^ 

This  reduces  the  equations  to 

(s  +  0'(«'-«')=8(s  +  0V2 («) 

and  2(s3_^3)^2  =26 (6) 

Divide  (a)  by  s  +  ^,  and  s  +  i  =  0,  or  s  =  —  «. 
Multiplying  out  the  quotient, 

s^-t^  +  st{s-t)=S^2. 
And  substituting  s^  —  t^  from  (5), 

s^(s-0=lV2   . (c) 

Dividing  (6)  by  (c), 

s^  ~  3  * 

Whence  i^-+il'  =  1?,  and  ^±-^  =  ±  2. 

.-.  5  =  3  «. 


SIMULTANEOUS   QUADRATICS.  173 

From  this  we  readily  obtain 

s  =  I V2,  t  =  1^2,  X  =  5,  y  =  4. 
Ex.  3.  Given    x(y  +  z)=  a,  y(z  +  x)=b,  z(x  +  y)=c. 
Adding  the  first  and  second  and  subtracting  the  third, 

xy  +  x»  -{-  yz  +  yx  —  zx  —  zy  =  2  xy  =  a  +  b  —  c. 
Similarly,         2  yz  =  b  +  c  —  a,  2  zx  =  c  +  a  —  b. 

Multiplying  together  two  of  these  new  equations  and  dividing 
by  the  third, 

2xy'2yz  _  2  ^^2  _  (o^  +  b  -  c)(b  +  c-  a) 
2zx  c-\-  a  —  b 


,^    lf(a  +  b-c)(b  +  c-  a)^ 
MX  2(c-\-a-b)  r 


2{c-\-  a-b) 
with  symmetrical  expressions  for  z  and  x. 

Ex.  4.   Given       x^  —  yz  =  a,  y^  ~  zx  =  b,  z^  —  xy  =  c. 
Then,  (x2  -  yz^  -  (2/2  -  zx)  {f  -  xy)  =  a^  -  be, 

i.e.    x(x^  +  y^  +  z^  —  S  xyz)  =  a^  —  be. 

,,^s^y3^,^_^^y^^a'-hc^b'^-ca^c'^-ab^ 
X  y  z 

since  the  left-hand  expression  is  symmetrical  in  ic,  2/,  and  z. 

Thence  each  fraction  ^  ^ |  (^^  -  ^^)^  "  (b'-  ^^)  (^'  "  ^^)  | . 

...  x  =  J\ a(a^-hcy I 

\  I  (a2  _  bey  -  (&2  _  ca)  (c2  -ab)r 

a^  -  be 


^(a^  +  b^-i-e^-Sabey 
with  symmetrical  expressions  for  y  and  z. 

EXERCISE  X.  d. 

1.  Given  x:y  =  S:2,  and  (2-xy  +  (l-  yy  =  25,  to  find  all 
the  values  of  x  and  y. 

2.  Given  x  +  ?/  =  a,  and  xy(x^  +  2/*^)  =  b. 


174  STMFLTANEOUS   QUADKATTCS. 

8.  Given  xy  =  750,  and  x  :  y  =  10  :  S. 

4.  Given  x  +  y  =  xy  =  x^  —  y'^ ;  that  is,  to  find  the  two  quanti- 
ties for  which  the  sum,  product,  and  difference  of  squares  may  be 
the  same. 

5.  Given  (x  -  y)  {x^  -  t/2)  =  160 ;  (x  +  ?/)  (x'^  +  y^)  =  580. 

6.  Given  x -\- y  -{-  xy  =  S4  ;  x"^  +  y^  -(x  +  y)=  42, 

7.  Given  x^ -\- y^ -\- x -\- y  =  SSO  ;  x'^  -  y^  +  x  -  y  =  150. 

8.  Given  4 x^  -\-  y"^  +  4x  +  2 y  =  6  ;  2xy  =  \. 

9.  Given  x  +  ?/  =  18  ;  x*  +  y*  =  14096. 

10.  Given  x  +  y  =  5  ;  (x^  +  ?/2)  {x^  +  y^)  =  455. 

11.  Given  -Vxn/  +  -V^Ti=  — • 

X  y  Z 

3 


12.  Given  ^ x_^x'^-y\^  ^_^+l^y. 

X     X  -\-  y  y         y         x        x 

13.  Given  x  —  Vx^  —  i/^  =  x(x  +  Vx^  —  y'^)  ; 

X  Vl  —  2/  =  y  Vl  +  X. 

14.  Given  xy  =  a  \  yz  =  h  )  zx  =  c. 

15.  Given  that  the  sides  of  a  right-angled  triangle  are  in  geo- 
metrical progression,  and  the  area  is  a^^  to  find  the  sides. 

16.  Find  three  numbers  such  that  the  product  of  each  into 
the  sum  of  the  other  two  may  be  the  numbers  48,  84,  and  90, 
respectively. 

17.  Given  x'^yH'^  =  a,  x^y^z"^  =  6,  x'^y'^z^  =  c,  to  find  x,  y,  and  z. 

18.  Giveny^=4:ax,  x~p=—2(a-\-x),  y  —  q=  ^.  (q  +  x)  to  find 
the  relation  between  a,  p,  and  q.  ^  ^ 

19.  Given  x2  +  xy  +  ^2  _  14  ^^  x^  -\- x^y^  +  y^  =  84  x2,  to  find  x 
and  y,     (Divide  one  equation  by  the  other.) 


SIMULTANEOUS   QUADRATICS.  175 

20.  x3  4-  2/3  +  xy{x  -{-y)  =  65,  (x^  +  y'^)x'"if  =  468,  to  find  x  and  y, 
(Put  x-\-  y  =  u  and  xy  =  v.) 

21.  x  +  y  +  z  =  lS,  ic2  4-  2/2  +  ;s2  _  ei,  jc?/  +  x^  =  2  2/^,  to  find 
X,  2/,  and  z. 

22.  The  sum  of  the  two  sides  of  a  right-angled  triangle  is  61, 
and  the  hypothenuse  is  greater  than  the  longer  side  by  3.  Find 
the  sides. 

23.  The  sum  of  the  three  sides  of  a  right-angled  triangle  is  60, 
and  the  sides  are  in  A.  P.     To  determine  the  triangle. 


CHAPTER  XI. 

Remainder  Theorem. — Tkansfoemation  of 
Functions. 

149.  When  we  divide  a^ -{- ax^ -{- bx -{- c  hyx—p,  we 
get  the  quotient  x^ -\-(a-\-p)x-\-p^ -^ap -\-b,  and  the 
remainder  p^  +  ap^  -{-bp  -{-c. 

We  notice  here  that  the  remainder  can  be  obtained 
from  the  dividend  by  simply  putting  p  for  x ;  or,  in  other 
words,  the  remainder  is  the  same  function  of  p  as  the 
dividend  is  of  x. 

To  prove  that  this  is  always  the  case. 

Let  fx  be  any  integral  function  of  x,  and  let  it  be 
divided  by  x—p.  Then  if  Q  denotes  the  quotient,  and 
B  the  remainder,  we  have 

fx  =  (x^p)Q-\-E, 

As  x—p  is  of  one  dimension  in  x,  E  is  independent 
of  X,  and  is  not  affected  by  any  change  in  x. 

Change  xtop-,  i.e.  put p  for  Xj  and  x—p  =  Oy  and 

which  proves  the 

Theorem.  If  an  integral  function  of  x  be  divided  by 
x  —  p,  the  remainder  is  the  same  function  of  p. 

Ex.  1.  The  remainder  when  ic^  —  5x*  +  6x  —  2  is  divided  by 
x-1  is  1»  -  5- 14  +  6. 1  -  2  =  0;  and  x  -  1  is  a  factor  of  the 
given  expression. 

176 


BEMAINDER   THEOREM.  177 

Ex.  2.   The  remainder  when  x^  —  6x^  +  Sx  —  2   is  divided  by 

ic  +  3  is 

(_  3)7  _  6(-  3)5  +  3(  -  3)  -  2  =  -  740. 

Ex.  3.  To  find  the  result  of  substituting  6  for  x  in  the  function 

x^-Sx^  +  x^-2x-l. 

To  substitute  6  for  x  is  to  find  the  remainder  when  the  function 
is  divided  by  x  -  6. 

Therefore,  6 

1_3_|_  i_  2-  1 
+  6  +  18  +  114  +  672 

1  +  3  +  19  +  112  +  671 
And  i?  =  671  is  the  result. 

Hence  to  substitute  a  for  x  in  an  integral  function  of 
X  is  equivalent  to  dividing  the  function  hj  x  —  a  and 
taking  the  remainder. 

EXERCISE  XI.  a. 

1.  Eind  the  value  of  x^'^  —  Sx"^  -\-  x^  —  bx  +  6  when  x  =  4,  when 
X  =  —  4,  when  x  =  l. 

2.  Find  the  value  of  x^  -  3.6  x'^  +  4.32  x  -  1.728  when  x  =  1.2. 

3.  What  is  the  remainder  when  x^  —  6  x*  +  5  x^  -  4  x'-^  +  3  x  —  2 
is  divided  by  x  +  5  ? 

4.  Find  the  remainder  when  (a  +  6  +  c)  (ab  +  6c  +  ca)  —  abc 
is  divided  by  a  +  6. 

5.  Find  the  remainder  when  x^  —  7  x  +  10  is  divided  by  x  —  1, 
by  X  -  2,  by  X  +  3. 

What  relation  does  x^  —  7  x  +  6  hold  to  the  three  divisors  ? 

6.  Find  the  result  of  substituting  1.71  for  x  in  the  function 

X8-'  5. 

What,  approximately,  is  the  relation  between  1.71  and  5  ? 


178  KEMAINDER   THEOREM. 

7.  Find  the  result  of  substituting  1.27  for  x  in  the  function 

What  relation  does  1.27  hold  to  the  function  ? 

8.  Find  the  remainder  when  x^  +  2  a:^  —  3  x^  +  x  +  1  is  divided 
by  5c2  -  X  +  1. 

The  function  may  be  written 

x(x*  +  2x2+  l)_  3x2  +  1, 

and  the  divisor  gives    x2  =  x  —  1. 

.-.  x{(x  -  1)2  +  2(x  _  1)+  l}~  3(x  -  1)+  1  =  i? 

=  x(x2)-3x+  4  =  x(x-l)-3x  +  4 

=  -3x  +  3. 

We  thus  substitute  x  —  1  for  x2,  wherever  x2  occurs,  and  con- 
tinue the  reduction  until  only  linear  x  remains. 

9.  What  is  the  value  of  x^^  -  x*  +  x  -  1  when  x3  +  2x— 1  =  0? 

150.  Divide  a^  — 3a^4-2aj  — 1  by  a;  — 1;  we  get  a 
quotient  ic^  —  2  a;  +  0,  and  a  remainder  —  1. 

Divide  a^  —  2ic4-0  byaj  —  1;  we  get  a  quotient  a;  —  1, 
and  a  remainder  ■—  1. 

Hence    a^-3a:2^2aj-l  =  (a:-l)(a^-2a;)-l 

=  (aj_l)(a;_l)(a;-l)  +  (x-l)(-l)-l 

We  have  thus  expressed  the  function  of  a;, 
a:3__3a;2^2aj-l, 
as  a  function  of  (aj  —  1),  viz., 

(a,_l)3_(a;_l)_l; 

and  we  notice  that  the  function  is  simplified  in  form  as 
it  lacks  the  square  term. 


REMAINDER   THEOREM.  179 

This  result  tells  us  that  to  substitute  any  value  for  x 
in  x^  —  3oc^  -{-2x  —  lis  equivalent  to  substituting  a  num- 
ber less  by  unity  in  the  function 

The  foregoing  transformation  may  also  be  effected  as 
follows : 

Let  x  —  l  =  y;  then  x  =  y  -\-l,  and  putting  y-\-l  for 
X  in  the  function,  and  expanding,  we  obtain 

f-y-^', 

or,  since  y  =  x  —  l,{x-'iy'-{x  —  l)^l, 

161.  Let  it  be  required  to  express  ic^  — 4a^  — 3a;  +  6 
as  a  function  of  x  —  2. 

The  transformed  function  will  take  the  form 

(x  -  2)3+  B2(x  -  2)2  +  E,{x  -2)-{-E, 

where  the  coefficients  i?,  Bi,  R^  ^^^  ^o  be  found. 

We  have  the  identity 

aj3__4aj2_3^^g^(^__2)3+i?2(^-2)2+i?,(a;-2)+i?. 

Writing  2  for  x  gives  ^  =  —  8. 

But  to  substitute  2  for  x  is  equivalent  to  dividing  by 
a;  —  2  and  taking  the  remainder ;  so  that  if  we  divide  the 
given  function  by  x  —  2  the  remainder  gives  R. 

Hence  rejecting  —  8  from  each  member,  and  dividing 
throughout  by  a;  —  2,  we  have  from  the  remainder 

The  next  similar  operation  gives 
R,  =  2. 


180 


EEMAINDER   THEOREM. 


Therefore  the  transformed  function  is 

(a;  _  2)3  +  2{x  -  2)2  -  7(x  -  2)  -  8. 
The  whole  operation  carried  out  in  the  form  of  divis- 

2 


ion  is  as  follows : 


■  4     -3 

2     -4 


+  '6 
-14 


1     -2 
+  2 

-7 
+  0 

-    8 

1     -0 

2 

-7  = 

=  .  .  . 

y     2  = 
1 


-B 


-B, 


^2 


We  here  divide  by  a;  ~  2,  getting  the  remainder  —  S=R, 

We  then  cut  this  off,  and  divide  the  quotient  of  —  2  a;  —  7 
by  X  —  2,  and  get  the  remainder  —  7  =  Ri.  Cutting  this 
off',  we  divide  again  by  a;  —  2,  etc.,  until  the  whole  is 
completed. 


Ex.  1.  To  express  x*  -  12x3  +  36x2  ■ 
x-3. 


•  X  —  77  as  a  function  of 


1-12 
3 

+  36 

-27 

-1-77 

+  27     +78 

1     -    9    .+    9 
3-18 

+  26 

-27 

+    1 

1-6-9 
3-9 

-    1 

1-3 
3 

-  18 

1           0 
And  the  required  functio 
(X  -  3)4  - 

n  is 

18  (x- 

_3)2_(a;-S 

3)+l. 


REMAINDER   THEOREM.  181 

Ex.  2.  To  express  m^  —  3  mhi  +  2  m/i^  —  Zn^  as  a  function  of 
m  —  n'm.  place  of  m. 

Both  parts  being  homogeneous,  this  is  equivalent  to  expressing 

/?»  y_8(''??V+2(™U3  as  a  function  of  ™  -  1. 
\n  I         \nj         \n)  n 

The  coefficients  are  readily  found  to  be  1,  0,  —  1,  —  3,  and  the 

function  becomes 

(m  —  nY  —  n^  (m  —  n)  —  3  n^. 

EXERCISE  XI.  b. 

1.  Express  x^  —  Zx'^-\-2x-\-l  as  a  function  of  sc  -f  1. 

2.  Express  Za^  —  a^-\-4iCfi  +  ^a  —  ^  as  a  function  of  &,  where 

3.  Express  ?/5  —  6  ?/*  +  10  y^  _  10  ?/2  +  5  ?/  —  2  as  a  function  of 
ic,  when  x  —  y  —  \. 

4.  Express  x^  +  ISx*  +  90^3  +  270  a:2  +  404x  +  241  as  a  func- 
tion of  X  +  3. 

5.  Express  a^-\-2a^h-2  a^b'^  -  12  a^b^  -  15  aS*  -  5  2)5  in  terms 
of  a  +  6  and  b, 

6.  Express  x^  —  7  x  +  6  as  a  function  of  sc  —  1. 

What  relation  does  x  —  1  hold  to  the  function? 

What  relation  does  1  hold  to  the  corresponding  equation? 

7.  If  x2  -^  ax  -\-  b  be  developed  as  a  function  of  x  —  z,  for  what 
value  of  z  will  the  function  take  the  form 

(x  -zy  +  A(x-z)? 

152.  If  we  have  to  express  fx  as  a  function  of 
x  —  {a-\-b-\-C'*'),  we  may  carry  out  the  operation  in 
successive  parts  by  putting  y  =  x  —  a,  and  transforming 
to  a   function  of  y ;  then  putting  z  =  y  —  h,  and  trans- 


182  BEMAINDER  THEOREM. 

forming  to  a  function  of  2,  etc.,  until  all  the  quantities 
a,  h,  c  •••  are  taken  in.  Where  6,  c,  etc.,  are  decimals, 
this  is  generally  the  most  convenient  method,  as  the 
whole  operation  is  very  compact. 

Ex.  To  express  x^  —  2  x^  +  3x  —  4  as  a  function  of  x  ~  1.23. 

In  the  following  operation  only  the  algebraical  sums  are  put 
down  in  each  column. 

1.23 


1-2 

+3 

-4         L 

-1 

+2 

-2 

0 

2 

1 

1.2 

2.24 

-1.552 

1.4 

2.52 

1.6 

1.63 

2.5689 

-1.474933 

1.66 

2.6187 

1.69 

Hence,   x^  -  2^2  4.  3^  -  4  =  (x  -  1)3  +  (x  -  1)2  +  2(x  -  1)  -  2 
=  (X  -  1.2)3  ^  i.6(x  -  1.2)2  _|_  2.52(x  -  1.2)  -  1.552 
=  (x  -  1.23)3  +  1.69(x  -  1.23)2  +  2.6187(x  -  1.23)  -  1.474933. 

The  work  may  be  still  more  condensed  by  leaving  out 
the  decimal  points,  and  remembering  that  when  begin- 
ning with  a  new  quotient  figure  the  product  must  be 
written  one  place  further  to  the  right  in  the  first  column, 
two  places  in  the  second  column,  three  in  the  third,  etc. 

153.   Let        /a;  =  a^-l-aj2-2aj-l  =  2/. 
When  ic  =  -  2,  - 1,  0,  1,  2,  ... 

2/  =  ~l,  +1,  -1,  -1,  +7,... 


BEMAINDEK   THEOREM. 


183 


The  graph  is  given 
in  the  figure,  with  0 
as  origin,  from  which 
values  of  x  are  meas- 
ured. 

Transform  to  a  func- 
tion of  z  where 

2;  =  a;  + 1, 
and  we  have 


2z'-z  +  l  =  y, 


And  when  2;  ==  —  1,  0,  1,  2,  3,  ••• 

2/  =  -l,  4-1,  -1,  -1,  -fJ... 

By  comparing  results  for  the  same  values  of  y,  we  per- 
ceive that  z  is  measured  from  an  origin  Z,  one  unit  to 
the  left  of  the  origin  for  x. 

Of  the  equation  ic^+ic^  —  2a;  —  1  =  0,  two  roots,  OP 
and  OQ,  are  negative,  and  one  root,  OB,  is  positive.  Of 
the  equation  2:^  —  22;^  —  2;  +  l  =  0,  one  root,  ZP,  is  nega- 
tive, and  two  roots,  ZQ  and  ZR^  are  positive. 

Similarly,  by  transforming  to  a  function  of  u  where 
u  =  x-\-2  =  z-{-lj  the  origin  is  moved  to  U,  and  the 
resulting  equation,  u^  ^  5u^  -{-6u-- 1  =  0,  has  all  its  roots 
positive. 

Hence  the  transforming  of  fx  to  a  function  of  a;  —  a  is 
equivalent  to  moving  the  origin  with  respect  to  the 
graph,  through  a  units  along  the  a;-axis,  to  the  right  if  a 
is  positive,  and  to  the  left  if  a  is  negative. 

Now  take  the  equation  a^  —  5a:^  +  6a;  — 1  =  0,  for 
which  the  origin  is  at  U,  and  all  the  roots  are  accordingly 
positive. 


184  KEMAINDER   THEOliEM. 

For  some  value  of  a,  the  origin  will  be  moved  from 
U  to  Pj  and  UP  represents  this  value  of  a.  But  UP 
also  represents  one  of  the  roots. 

Therefore  the  value  of  a  which  transfers  the  origin 
from  U'  to  P  is  one  of  the  roots  of  the  equation. 

Similarly,  the  values  of  a  which  transfer  the  origin 
from  Uio  Q,  and  from  Uto  R,  are  the  tw^o  other  roots.    . 

154.  As  the  roots  of  the  equation  x^  —  bx^-\-^x  —  l=zO 
are  real  and  incommensurable,  they  may  be  approximated 
to.     Let  us  then  endeavor  to  find  the  value  of  UR. 

For  this  purpose  we  first  transfer  the  origin  through 
three  units  from  Uto  F.  Then,  having  carefully  drawn 
the  graph,  we  estimate  the  distance  VR  in  tenths  of  a 
unit,  as  nearly  as  we  can,  and  we  move  the  origin 
onwards  through  this  estimated  distance. 

To  know  whether  our  estimated  distance  is  too  great 
or  not,  we  have  the  following  test  ; 

As  long  as  the  origin  lies  between  Fand  R,  the  graph 
at  that  point  is  below  the  ic-axis,  and  the  independent 
term  is  negative  ;  but  when  the  origin  passes  R,  the 
graph  rises  above  the  a;-axis,  and  the  independent  term 
is  positive. 

Hence  a  change  in  the  sign  of  the  independent  term 
indicates  that  we  have  caused  the  origin  to  pass  R, 

Now  transforming  cc^  —  5a^4-6aj  —  1  =  0  to  a  function 
of  a;  — 3  gives  ?/^4-4?/2  + 3i/  — 1,  where  y  =  x^S. 

Next  transform  to  a  function  oi  y  —  0.2,  and  we  get 
z'  +  4:.6z''-i-4.,72z- 0,232,  where  z  =  y -0.2',  and  the 
independent  term  being  —  shows  that  our  new  origin  is 
still  to  the  left  of  R. 


REMAINDER   THEOREM.  185 

The  student  is  advised  to  try  the  effect  upon  the 
independent  term  of  transforming  to  y  —  0.3. 

Again  transforming  the  last  equation  to  a  function 
of  z-OM,  we  obtain  u^ +  4.72  u^ -{- 5.0928  u- 0.035776, 
where  n  =  z  —  0.04. 

We  have  thus  transferred  the  origin  in  all  through 
the  distance  3.24,  and  this  is  to  two  decimals  a  correct 
approximation  to  the  root  Uli.  A  repetition  of  the 
same  process  will  furnish  as  close  an  approximation  as 
may  be  desired. 

The  work  is  carried  out  practically  in  the  following 
condensed  form,  where  only  algebraic  sums  are  written, 
and  decimal  points  are  not  employed : 

3.24698 


1-5 

+  6 

-1 

-2 

0 

-1... 

+  1 

3:. 

-0232... 

4. 

384 

-0035776... 

42 

472.. 

-0005047984 

44 

49056 

46. 

50928.. 

464 

5121336 

468 

5149728 

472. 

4726 

4732 

4738 

It  will  be  noticed  that  the  independent  term  is  being 
successively  reduced  in  value  ;  and  as  this  term  gives  the 
distance  from  the  origin  to  the  graph,  measured  parallel 
to  the  ?/-axis,  this  reduction  shows  that  the  origin  is  ap- 
proaching the  point  R. 

We  have  said  that  the  first  decimal  figure  must  be 
estimated,  and  then  tried.     So,  to  a  certain  extent,  must 


186  REMAINDER   THEOREM. 

the  remaining  figures.  But  the  values  of  the  others  are 
readily  found.  Thus,  after  finding  the  2  of  the  quotient, 
and  finishing  the  transformation,  we  add  a  cipher  to  0232 
of  the  last  column,  and  divide  by  472  of  the  second  col- 
umn ;  this  gives  4  for  the  next  figure.  Similarly,  357760 
divided  by  50928  gives  6  for  the  next,  and  so  on. 

In  fact,  after  obtaining  the  three  decimals  246,  and 
completing  the  transformations  thus  far,  we  may  safely 
obtain  three  more  decimals  by  simple  division.  In  this 
way  9,  8  are  obtained  by  dividing  5047984  by  514972. 

Ex.  To  find  the  root  UQ.  The  various  transformations  give 
as  coefficients  — 

for  X  -  1,  1?  -  2,        ~  1,  +  1  ; 

for  X  -  1.5,  1,   -  0.5,     -  2.25,       +  0.125  ; 

for  x-1.55,  1,   -0.35,   -2.2925,   +0.011375; 

and  the  approximate  root  is  1.554  ••• 

It  will  be  noticed  that  the  independent  term  for  this  root  is  +, 
as  the  graph  lies  above  the  x-axis  to  the  left  of  §,  or  between  P 
and  Q, 

155.  The  preceding  methods  offer  an  elegant  means  of 
extracting  roots  of  numbers. 

Ex.  To  approximate  to  the  cube  root  of  12.  Let  x^  —  12  =  0, 
and  solve  this  as  a  cubic  equation.  This  equation  has  but  one  real 
root,  and  that  is  the  arithmetic  cube  root  of  12. 


0 

0 

-12    12.2894 

2 

4 

-  4 

4 

12.. 

-  1.352... 

6 

1324 

-  0.147648 

62 

1452.. 

64 

150544 

66 

155952 

668 

676 

•.  ^2  =  2.2894... 

REMAINDER   THEOREM.  187 


EXERCISE  XI.  c. 

1.  Determine  the  integral  values  between  which  the  real  roots 
of  the  following  equations  lie  — 

i.   x^-Sx^-\-2x-2.  iii.   x*  -  4x2  +  3a;  -  4. 

ii.   x3+3x2  +  2x  +  2.  iv.   x*  +  2x3-4x-2 

•      2.  Find  to  3  decimals  the  greatest  positive  root  of 
x4_  2x3-3x2  +  6x-l  =0. 

3.  Transform  x^  —  3  x^  +  3  x  —  4  =  0  to  an  equation  in  (x  —  1), 
and  thence  find  the  roots  of  the  given  equation. 

4.  Transform  x*  —  4 x^ +  2 x- +  4x4-6  =  0  to  an  equation  in 
(x  —  1),  and  thence  find  the  roots  of  the  given  equation. 

5.  Show  that  if  x^  —  px^ -]- qx  +  r  =  0  be  transformed  to  an 
equation  in  (x— ^),  the  equation  will  assume  the  form  z^+  Qz 
4-  i?  =  0,  where  the  square  term  is  wanting. 

6.  Remove  the  second  term  from  x^  —  Gx^  +  12x  +  9  =  0,  and 
thence  solve  the  equation. 

7.  In  the  equation  x^  —  2  x^  —  x  —  6  =  0,  move  the  origin  3  units 
to  the  right,  and  thence  find  the  roots. 

8.  Find  to  5  decimals  the  cube  root  of  3.1416. 

9.  A  gallon  contains  277.273  cu.  in.  Find  the  length  in  inches 
of  the  edge  of  a  cubical  box  that  shall  hold  just  10  gallons. 

10.  Find  the  fifth  root  of  100,  to  3  decimals. 


CHAPTER   XII. 
The  Progressions.  —  Interest  and  Annuities. 

m 

156.  A  series  is  a  succession  of  terms  which  follow 
some  fixed  law,  by  means  of  which  any  term,  after  some 
fixed  term,  usually  not  far  removed  from  the  beginning, 
may  be  obtained  from  the  preceding  terms  and  from 
constants. 

Thus  l-{-3  -\-  5  -\-7  -\-  "'  is  a  series  in  which  each 
term  is  got  from  the  preceding  one  by  adding  2. 
1  +  i  +  i  +  i'"  ^^  ^  series  in  which  each  term  is  one- 
half  the  preceding  term. 

The  doctrine  of  series  is  a  very  extensive  and  im- 
portant one,  and  has  given  rise  to  a  distinct  calculus, 
that  of  Finite  Differences ;  but  two  series,  the  simplest 
of  their  species,  are  of  such  common  application  as  to  be 
treated  of  in  elementary  algebra,  and  even  in  arithmetic, 
under  the  name  of  the  Progressions. 

A  series  in  which  each  term  differs  from  the  preceding 
one  by  a  constant,  as  in  the  first  of  the  foregoing  exam- 
ples, is  an  Arithmetic  Series,  or  an  Arithmetic  Progres- 
sion, and  is  symbolized  as  an  A.  P. ;  and  a  series  in  which 
each  term  is  a  constant  multiple  of  the  preceding  one, 
as  in  the  second  example,  is  a  Geometric  Series,  or  a 
Geometric  Progression,  contracted  to  G.  P. 

A  third  kind,  the  nature  of  which  will  be  explained  in 
the   proper   place,  is   called   a   Harmonic    Series,  or   a 
Harmonic  Progression,  contracted  to  H.  P. 
188 


ARITHMETIC   SERIES.  189 


ARITHMETIC    SERIES. 

157.  The  quantities  normally  occurring  here  are :  a, 
the  first  term  of  the  series ;  d,  the  common  difference ; 
w,  any  given  number  of  terms  ;  and  s,  the  sum  of  n 
terms. 

If  fn  is  such  a  function  of  n  that  the  substituting  of 
any  integral  number  for  n  gives  that  numbered  term  in 
the  series,  fn  is  called  the  7ith  term  of  the  series,  and 
is  all-important,  not  only  in  an  A.  P.,  but  in  all  series, 
as  expressing  the  law  of  the  series. 

Evidently,  to  know  the  form  of  fn  is  to  know  the  series, 
since  its  consecutive  terms  are  given  by  the  substitution 
of  1,  2,  3,  •  •  •  etc.,  for  n. 

The  consecutive  terms  of  an  A.  P.  are 

a,  a  +  cZ,  a -{-2d,  a  +  3  d,  etc., 

and  it  is  readily  seen  that  the  nth  term  is  a-{-(7i  —  l)  d. 
It  being  often  convenient  to  denote  this  general  term  by 
a  single  letter,  z,  we  have 

z  =  a-{-(n  —  l)d (A) 

158.  An  A.  P.,  like  any  other  series  of  numbers,  is  not 
necessarily  limited  in  extent,  but  may  be  continued  at 
pleasure  in  either  direction. 

When  we  consider  any  portion  containing  n  consecu- 
tive terms  of  this  unlimited  series,  we  call  the  first  term 
a,  and  the  nth,  or  last  term,  z.  And  thus  any  two 
unequal  numbers  may  be  any  two  terms  of  an  A.  P. 

Ex.  To  find  the  A.  P.  whose  5th  term  is  12,  and  whose  11th 
term  is  24. 


190  ARITHMETIC    SERIES. 

There  will  evidently  be  7  terms  in  this  portion,  and  hence 

24  =  12  +  6d 

.  •.  d  =  2;  and  the  first  term  of  the  series  is  12  —  4  df  or  4. 
Hence  the  nth  term  is  4  +  2(/i  —  1)  or  2  +  2  n.    And  the  series  is 

4  +  6  +  8+10+  — 

159.  It  will  be  noticed  that  the  nth  term  of  an  A.  P. 
is  a  linear  function  of  n. 

Now  any  function  of  n  taken  as  the  nth  term  will 
give  rise  to  some  series,  and  if  it  is  a  positive  integral 
function  of  n  higher  than  linear,  the  series  will  be  of 
the  same  species  as  an  A.  P.  but  of  a  higher  order.  So 
that  the  A.  P.  is  the  simplest  series  of  its  species. 

Thus  if  the  nth  term  be  l(n^-\-7i)  or  ^n(n -f  1)?  the 
series  is 

1,  3,  6,  10,  15  ... 

It  is  worthy  of  notice  that  the  differences  of  the  terms 
in  this  series  form  an  A.  P. 

Similarly,  if  the  nth  term  be  of  the  form  n^  +  n^,  or  a 
function  of  three  dimensions  in  n,  the  differences  of  the 
differences  of  the  terms  of  the  series  will  be  an  A.  P. 

160.  As  n  denotes  the  number  of  a  term  in  a  series,  it 
must  be  integral ;  and  hence  the  presentation  of  a  non- 
integral  value  for  n  indicates  some  absurdity  or  im- 
possibility. 

Ex.  Is  100  a  term  of  the  series  whose  first  term  is  3  and  whose 
difference  is  4  ? 

As  the  nth  term  is  3+(w  — 1)4  or  4n  — 1,  the  equation 
100  =  4  71  —  1  will  give  an  integral  value  for  n  if  100  is  a  term  of 
the  series. 


ARITHMETIC    SERIES.  191 

It  does  not  give  such  a  value,  and  therefore  100  is  not  a  term  of 
the  series. 

It  is  readily  seen  that  99  is  the  25th  term. 

161.   S  being  the  sum  of  n  terms  of  an  A.  P.,  we  have 


^  =  a  +  (a  +  d)  +  (a  +  2d)H (a  +  7i  — l-d). 

Also,  by  reversing  the  order, 


S=(a-\-n  —  l'd)-{-(a+n—2'd)-j-(a  +  n—3'd)-] ha. 

Adding, 


2S  =  (2a  +  n  —  l'd)-\-  (2a  +  n  —  l'd)-] n  terms 

=  n(2a  +  n  —  l'd). 


.'.  S=~(2a  +  n-l'd) (B) 

The  following  method  of  investigation  for  the  sum  of 
an  A.  P.  is  important. 

We  have         1^  =  02  +2-0         +1, 

2^=12  +2.1  +1, 

32  =  22  +2.2  +1, 


n^  =  {n  -\y  -\-2{n  -\)  ^\ 

(7l  +  l)2=n2  +271  +1. 

.-.  by  addition, 

2(l  +  2  +  3+...n)  =  (n  +  l)^-(n  +  l); 

whence  S  =  — i — ~^ — ^  =  the  sum  of  the  first  n  natural 

numbers. 

Then,  the  terms  of  an  A.  P.  are 

a,    a  +  d,    a-\-2d^    ...    a-\-{n  —  V)d\ 


192  ARITHMETIC    SERIES. 


and  summing  these  gives 


>S'=  na -h  (1  +  2  4- 3  +  ...  n  -  l)cZ 
=  na-{-  ^n(n  —  l)d. 
And  this  is  the  same  as  (B), 
Ex.   The  sum  of  the  first  n  odd  numbers  is 
^(2  +  5r^ri.2),  orn2. 

162.  Upon  multiplying  out, 

fv        2^1        2a  —  d 

JS  =71"^ \-  n  ' .• 

2  2 

Hence  the  sum  of  n  terms  of  an  A.  P.  is  a  quadratic 
function  of  n,  with  no  independent  term. 

And  every  quadratic  in  n,  without  the  independent 
term,  is  the  sum  of  ?i  terms  of  some  A.  P.  The  inde- 
pendent term,  if  present,  would  appear  as  an  extraneous 
term  which  might  or  might  not  follow  the  law  of  the 
series. 

Ex.  To  find  the  A.  P.  of  which  2  n^  —  3  n  expresses  the  sum  of 
n  terms. 

Let  n  =  I  ;  the  sum  of  1  term  is  —  1  =  «. 

Let  n  =  2  ;  the  sum  of  2  terms  is2  =  2a  -{-  d. 

.'.  d  =  i,  and  the  7ith  term  =  4  w  —  5  ; 
which  gives  the  series. 

Otherwise,  the  sum  of  n  terms,  or  Sn,  is  2n^  —  Sn;  and  the 
sum  of  w  —  1  terms,  or  Sn-h  is2(n  —  ly  —  3(w  —  1). 

But  Sn  is  got  by  adding  the  nth  term  to  Sn-i. 

.-.  nth  ieYm= Sn- >Sn-i=2  n^-S n-2(n-iy-{-S(n-l)=4:n-6, 

163.  Any  positive  integral  function  of  ?i,  of  a  higher 
degree   than  the  second,  and  lacking  the   independent 


ARITHMETIC   SERIES.  193 

term,  expresses  the  sum  of  n  terms  of  some  series  of  the 
same  species  as  the  A.  P.  but  of  a  higher  order. 

Ex.   Let  2  n^  —  3  n^  +  n  be  the  sum  of  n  terms. 
Then  Sn-Sn-i=^n^-Sn^+n  -  2(n-iy+S(n-iy-{n-'i) 
=  6  n2  -  12  ?i  +  6, 
which  is  the  nth  term.    And  the  series  is 

0  +  6  +  24  +  54  +  ..., 
a  series  whose  differences  form  an  A.  P.     Compare  Art.  159. 

164.  Tn  problems  where  n  is  to  be  found  from  con- 
ditions involving  the  sum  of  n  terms,  n  may  have  two 
values.  If  both  be  integral,  both  will  satisfy  the  con- 
ditions, but  non-integral  values  of  n  must  be  rejected  as 
being  inapplicable  to  the  case. 

Ex.  How  many  terms  of  the  series  whose  nth  term  is  27  —  2  n 
will  make  144  ? 


The  sum  of  n  terms  is  -  (2  a  +  n  —  1  •  d),  and  this  is  to  be  144. 
We  readily  find  a  =  25,  and  d=  —  2, 

.-.  ^(52-2n)=144.  "^ 

Whence  n  =  18,  or  8. 

And  the  sum  of  18  terms  =  sum  of  8  terms  =  144. 

165.  When  three  quantities  form  three  consecutive 
terms  of  an  A.  P.,  the  middle  one  is  called  an  arithmetic 
mean  between  the  other  two. 

Let  A  be  an  arithmetic  mean  between  a  and  b.  Then 
A  —  a  =  b  —  A]  whence 

A  =  ^{a  +  b). 


194  ARITHMETIC   SERIES. 

Or  the  arithmetic  mean  between  two  quantities  is  one- 
half  their  sum. 

The  following  miscellaneous  examples  illustrate  the 
subject  of  arithmetic  progression. 

Ex.  1.  The  pth  term  of  an  A.  P.  is  P  and  the  qth  term  is  Q,  to 
find  the  nth  term,  and  the  sum  of  n  terms. 

From  the  pth  to  the  ^th  term  the  difference  is  added  q—p 
times. 

q-p 

Also,  from  the  first  to  the  pth  term  the  difference  is  added  p—1 
times. 

,.a  =  P-(p-l).Q^z^=lil^L}l:^QLP^zJ}. 
q-p  q-p 

Hence  the  nth  term  is  P^-QP+<Q-P). 

q-p 

Also,  s=-(ia-\-  71th.  term) 


_nf2(Pq-Qp)  +  (n+l)(Q-P} 
21  q-p  } 


Ex.  2.  In  the  A.  P.'s  6  +  7^  +  9  +  .••   and  -  3  -  1  +  1 ...   to 
determine  — 

(1)  If  there  be  a  common  term,  and  if  so,  its  value. 

(2)  If  there  be  a  common  number  of  terms  for  which  the  sum 
in  each  series  is  the  same,  and  if  so,  to  find  the  simi. 

(1)  6  +  (w  —  1)1  =  —  3  +  (n  —  1)2  gives  an  integral  value  for  n 
if  there  is  a  common  term. 

This  gives  n  =  19,  and  the  19th  term  is  common. 
Its  value  is  6  +  18  X  I  or  33. 

(2)  /S'  =  ^[l2  +  ^TTT.  ?^  =  ^  (-  6  +  W^^l .  2)  gives  the  condi- 
tion for  a  common  sum,  if  n  is  integral 


ARITHMETIC    SERIES.  195 

This  gives  n  =  37,  and  the  sum  of  the  first  37  terms  is  the  same 
in  each  series. 

The  sum  is  V-  (12  +  36  x  f)  =  1221. 

EXERCISE  XII.  a. 

1.  Find  the  nth  terms  in  the  A.P.s,  two  of  whose  terms  are 
given  as  follows  — 

i.  1st  term  =  wi,  2d  =  ^.  v.  3d  =  8,  8th  =  6. 

ii.  1st  =  a  +  n-l  -6,  2d  =  a.  vi.  10th  =  4,  4th  =  20. 

iii.  1st  =  3,  3d  =  12.  vii.  (w-l)th=a,  (7H-l)th=&. 

iv.  1st  =  0,  10th  =  60. 

2.  Sum  the  following  A.  P.s  — 

1.  1  +  IJ  +  •••  to  12  terms.  iv.  -  H h  •••  to  n  terms. 

a     a-\-  b 

ii.  i  +  i  +  —  to  10  terms.  v.   103  +  97  +  •••  to  24  terms, 

iii.  0  +  3  +  .-.  to  7  terms.  vi.  2  +  4  +  •••  to  n  terms. 

8.   Sum  100  terms  of  the  A.  P.  whose  3d  term  is  5,  and  10th 
term  75. 

4.  Sum  to  n  terms  the  series  whose  nth  term  is  J  (n  —  1). 

5.  Find  the  sum  of  all  the  multiples  of  7  lying  between  200  and 
400. 

6.  If  a,  6,  c  are  in  A.  P.,  so  also  are  a^  (6  +  c),  b^(c  +  a),  and 
c2(a  +  6). 

7.  Find  the  A.  P.  for  which  s  =  }  (3  ^2  -  2  m). 

8.  Find  the  A.  P.  for  which  s  =  72,  a  =  17,  d  =  -2. 

9.  The  A.  P.s  whose  sums  are  29  n  —  2  n^   and  J  (17  n  +  3  n^) 
have  a  common  term.     Find  it. 

10.  One  hundred  apples  are  placed  in  line  at  two  feet  apart, 
and  a  basket  is  placed  at  an  extreme  apple.  How  far  will  a  person 
travel  who  takes  the  apples  one  by  one  to  the  basket  ? 


196  ARITHMETIC    SERIES. 

11.  n  apples  are  placed  in  line  d  feet  apart,  and  a  basket  is 
placed  in  the  same  line,  m  feet  from  the  first  apple.  How  far  will 
a  person  travel  who  takes  the  apples  one  by  one  to  the  basket  ? 

What  is  the  difference  between  m  positive  and  in  negative  ? 

12.  A  debt  of  •$  1000  is  paid  in  20  annual  instalments  of  $  50 
with  simple  interest  at  6%  on  all,  due  at  the  time  of  each  pay- 
ment.    How  much  money  is  paid  in  discharging  the  debt  ? 

13.  A  person  receives  ^1.50  daily.  He  spends  15  cents  the  first 
day,  18  the  second,  21  the  third,  and  so  on. 

i.   When  will  he  be  worth  the  most,  and  how  much  will  it  be  ? 

ii.   When  will  he  be  worth  nothing  ? 
iii.   When  will  he  be  worth  exactly  $  21.30  ? 
iv.   Will  he  at  any  time  have  exactly  1 10  ? 

14.  A  and  B  start  from  the  same  place  at  the  same  time.  A 
goes  westward  10  miles  the  first  day,  8  miles  the  2d,  etc.,  in  A.  P. 
B  goes  eastward  3  miles  the  1st  day,  4  miles  the  2d,  etc.,  in  A.  P. 

i.   Where  and  when  will  they  be  together  ? 
ii.   When  will  they  be  70  miles  apart  ? 

15.  Eind  the  sum  of  1  +  2  -  3  +  4  -  5  +  6  -  +  •••  to  120  terms. 

16.  Two  sides  of  an  equilateral  triangle  are  each  divided  into 
100  equal  parts,  and  corresponding  points  of  division  are  joined. 
Find  the  total  length  of  all  these  joins. 

Into  how  many  equal  parts  must  the  sides  be  divided,  so  that 
the  sum  of  the  joins  may  be  m  times  the  side  of  the  triangle  ? 

17.  $  100  is  deposited  annually  in  a  bank  for  20  years  to  be  left 
at  simple  interest  at  6  %.  What  is  the  accumulated  sum  at  the 
last  payment  ? 

18.  The  side  of  an  isosceles  triangle  is  a  and  the  base  is  &,  and 
the  altitude  is  an  arithmetic  mean  between  the  side  and  base. 
Show  that  a  :  6  =  1  +  ^7  :  3. 

19.  Insert  4  terms  of  an  A.  P.  between  a  and  b. 


GEOMETRIC   PROGRESSION.  197 

20.  Find  the  series  for  which  S  =  in(n  -\-  1) (n  +  2). 

21.  Find  the  series  for  which  S  =  ^  n  (n  -\-  1)  (2  n  +  1). 

22.  Find  the  series  for  which  S  =  {in(n  +  l)}^. 

23.  The  sums  of  two  A.  P.s  to  n  terms  each,  are  n^  +  pn  and 
3  w^  —  2  n.  For  what  value  of  p  will  they  have  a  common  nth 
term  ? 

Show  that  they  cannot  have  a  common  term  unless  j)  is  a  mul- 
tiple of  4. 

24.  Two  sets  of  n  lines  each  are  drawn  parallel  to  adjacent  sides 
of  a  parallelogram.  Find  the  whole  number  of  parallelograms  thus 
formed. 

26.  The  natural  numbers  are  divided  into  groups  as  follows  — 

(1)(2,3)(4,  6,6)(7,  8,  9,  10)... 
Find  the  sum  of  the  numbers  in  the  nth  group. 

GEOMETRIC    PROGRESSION. 

166.  The  quantities  with  which  we  have  normally  to 
deal  in  Geometric  Progression  are 

a,  the  first  term  ;  r,  the  common  ratio ;  ?i,  the  number 
of  terms ;  z,  the  last  or  nth  term ;  and  JS,  the  sum  of  n 
terms. 

By  definition,  the  terms  of  a  Geometric  Progression  are 

a,  ar,  ar^,  ar^  •••,  and  it  is  readily  seen  that  the  nth  term 

is  af~^, 

.-.  a;  =  ar"-^ {A) 

Again,  8  ==  a -{-  ar  -\-  ar^  -{-  •  •  •  a?*"~\ 

and  rS  —         ar  -^an^  -\ ar""'^  +  ar^ 

...  ^(i_r)=a(l-r«), 

and  S^a^'^^ (J5) 

1  —  r 


198  GEOMETRIC   PROGRESSION. 

Eelation  (B)  may  also  be  obtained  as  follows : 
By  division, 

-  =  a  +  ar  +  ar^  +  •  •  •  ar"~^  + 


1  —  r  1  —  7' 

I         1^1  n-i         ^  ^^"  1  —  r" 

1  —  r      1  —  r  1 —r 

Ex.  The  population  of  a  city  increases  at  the  rate  of  5%  per 
annum,  and  it  is  now  20000.     What  will  it  be  10  years  hence  ? 

It  will  evidently  be  the  11th  term  of  the  G.  P.  whose  first  term 
is  20000,  and  whose  ratio  is  1.05. 

.-.  The  population  will  be  20000  (1.05)io  =  32578  to  the  nearest 
Integer. 

167.  The  finding  of  r  or  n  in  a  geometric  progression 
cannot  usually  be  conveniently  done  without  employing 
logarithms.  Thus  in  the  preceding  example  the  labor 
of  raising  1.05  to  the  10th  power  is  very  great.  And  in 
the  case  of  n,  on  account  of  its  appearing  as  an  exponent, 
the  common  operations  of  arithmetic  do  not  suffice  for 
finding  it. 

Ex.  1.  In  a  G.P.  the  first  term  is  J,  and  the  second  term  is  i; 
to  find  the  wth  term  and  the  sum  of  n  terms. 
Since  a  =  \  and  ar  =  ^.    .-.  r  =  J  -;- 1  =  |. 

Then  z  =  nth  term  =  -  {-Y~^  =  — !• 
3V2/  2«-i 

Ex.  2.  How  many  terms  of  the  series  l  +  2  +  4  +  8  +  »»»  will 
make  127  ? 

Here  m  ='^^^^^  =  '^^^^^:^  =  2^  -  1. 

r-\  1 

.•.  2»  =  128,  and  n  is  evidently  7. 


GEOMETRIC   PROGRESSION.  199 

168.  When  r  <  1,  r"  diminislies  as  n  increases,  and  by 
taking  n  great  enough  r"  may  be  made  as  small  as  we 
please.    At  the  limit  when  n  =  oo ,  r"=0,  and  {B)  becomes 


1-r 


This  is  the  limit  towards  which  the  sum  of  n  terms  of  the 
series  approaches  as  n  is  continually  increased ;  and  by 
taking  n  sufficiently  great  we  can  make  the  sum  of  n 
terms  approach  this  value  as  near  as  we  please. 

For  convenience,  this  limit  is  called  the  sum  of  the 
series  to  infinity,  although  it  is  more  properly  spoken  of 
as  the  limit  of  the  series  as  n  approaches  infinity. 

Ex.  1.  To  find  the  limit  of  0.333  *"  ad  infinitum. 


And  its  limit  is  A/i  +  J_-|.  JL  +  ...\  =  A 1_=1. 

10  V        10     102         I     10  l-yi^j     3 

Ex.  2.  To  find  the  value  of  the  circulating  decimal  0.241. 

10      103  V    ^102^104^      I      10     103    99 

_  2        41  ^  2  X  99  +  41  ^  241  -  2 
10      990  990  990 

Hence  the  rule  —  Subtract  the  non-repeating  part  from 
the  whole,  and  write  as  denominator  as  many  9's  as 
there  are  digits  in  the  repeating  part,  followed  by  as 
many  ciphers  as  there  are  digits  in  the  non-repeating 
part. 


200  GEOMETKIC   PROGRESSION. 

EXERCISE   XII.   b. 

1.  Find  the  sum  of  n  terms  of  the  series  — 

i.    1  +  3  +  9  +  ...  ii.    i_|  +  |-_^.. 

iii.  y3^  +  yf  ^  +  •••     What  is  this  when  w  =  oo  ? 
iv.    ^2  +  (2  —  ^2) 4-  -•  when  n  =  co. 

2.  Sum  to  00  the  series  1  H \-       ^         ' 


n  +  1      (n  +  1)2 
What  are  the  results  when  n  =  l?    =2?    =S? 

3.  Find  the  nth.  term  of  the  G.  P.  whose  first  term  is  v    + 

1  V2-1 

and  the  second  is 

2-V2 

4.  A  cask  of  wine  contains  30  gallons.  5  gallons  are  drawn  off 
and  the  cask  is  filled  up  with  water.-  After  this  has  been  done 
6  times,  how  many  gallons  of  the  original  wine  are  in  the  cask  ? 

5.  A  circle  is  inscribed  in  an  equilateral  triangle ;  a  second 
circle  touching  the  first  and  two  sides  ;  a  third  touching  the  second 
and  the  same  two  sides,  etc.  If  the  side  be  s,  find  the  radius  of 
the  nth  circle  so  described.  Also,  find  the  total  area  of  all  the 
circles  continued  to  oo. 

6.  A  square  is  inscribed  in  an  acute-angled  triangle  having  one 
side  of  the  square  on  the  base  of  the  triangle.  A  second  square  is 
inscribed  similarly  in  the  triangle  above ;  a  third  square  above 
that ;  etc.  to  x.  If  the  base  of  the  triangle  be  b  and  its  altitude  a, 
find  the  total  area  of  all  the  squares. 

7.  How  many  terms  of  the  G.  P.  1  +  2  +  4  +  ...  will  make  1023  ? 

8.  Find  the  sum  of  n  terms  of  the  series  whose  nth  term  is 
na  +  a\ 

9.  If  S,  a,  r,  z  be  taken  in  their  common  acceptation,  show  that 

r  ■= 

S-z 

10.   A  country  whose  annual  production  is  now  5  millions,  in- 
creases at  the  rate  of  5  %  per  annum.    What  will  it  be  5  years  hence? 


GEOMETRIC   PROGRESSION.  201 

169.  When  three  quantities  are  three  consecutive 
terms  of  a  G.  P.,  the  middle  quantity,  is  a  geometric  mean 
between  the  other  two.  It  is  also  called  a  mean  propor- 
tional.    Art.  84. 

Let  (7  be  a  geometric  mean  between  a  and  h. 

Then  —  =  — - :  and  hence  G  =  Va6. 

a      G 

Problem.     To  insert  n  terms  betwen  two  extremes,  a 
and  h,  so  that  the  whole  may  be  a  G.  P. 
Let  the  series  be 

Then  ^-l  =  ^2^...^  =  A.=  ^  =  ,.. 

a     t^  t,_,     ^„_i      t 

and  ^  =  ^.^...^.  A..^  =  ^.n+i, 

«       Cf'    h       tn-2    L^i    K  ' 

.-.  r  =  [-Wi;  ^i  =  ar  =  (a"6)»+i; 

1  1 

e2  =  (a'»-^62)«+^;  and  generally,  «^  =  (a'»-*"+^6'")"+^ 


EXERCISE  XII.  c. 

1.  Insert  8  terms  between  1  and  512,  to  form  a  G.  P. 

2.  A  body  weighs  a  grams  in  one  scale  pan  of  a  balance  and  h 
grams  in  the  other.     Show  that  its  true  weight  is  Vab, 

3.  Three  circles  each  touch  the  same  two  lines,  and  one  circle 
touches  both  the  others.  Show  that  the  radii  of  the  circles  form 
aG.P 

4.  If  a,  6,  c,  d  be  in  G.  P.,  prove  that 

(a  +  6  +  c  +  d)2  =(a  +  b)2 +(c  +  d)'^  +  2(6  +  c)2. 


202  HARMONIC   SERIES. 

5.  A  right-angled  non-isosceles  triangle  has  a  perpendicular 
drawn  from  the  right-angled  vertex  to  the  hypothenuse.  The 
larger  of  the  resulting  triangles  is  treated  in  the  same  manner; 
and  so  on.     Show  that  the  perpendiculars  so  drawn  form  a  G.  P. 

6.  AB  is  the  diameter  of  a  circle,  and  CT  is  a  tangent  at  any 
point  C  on  the  circle,  and  AT  is  perpendicular  to  CT.  Prove  that 
AC  is  Si  geometric  mean  between  AB  and  AT. 


HARMONIC    SERIES. 

170.  A  number  of  terms  form  a  Harmonic  Series  when 
their  reciprocals  form  an  Arithmetic  Series. 

Thus  1,  2,  3,  4,  5,  etc.,  are  in  A.  P. 

And  1,  I,  ^,  \,  ^,  etc.,  are  in  H.P. 

Let  a,  by  c  be  three  terms  in  H.  P. 

Then  i,  ^,  i  are  in  A.  P.,  and  ?  =  -  +  31:. 

a   b    c  b     a     c 

a_^a  —  b 
c      b  —  c 

That  is,  three  quantities  are  in  H.  P.  when  the  first  is  to 
the  third  as  the  difference  between  the  first  and  second 
is  to  the  difference  between  the  second  and  third. 

The  term  Harmonic  is  derived  from  the  property  that 
a  string  of  a  musical  instrument  stopped  at  lengths  cor- 
responding to  the  terms  of  an  H.  P.,  sounds  the  harmonics 
in  music. 

In  algebra  itself  Harmonic  Progression  does  not  play 
any  important  part;  it  is  in  geometry  that  it  has  its 
principal  applications. 

JS"o  method  of  summing  an  H.  P.  is  known. 


INTEREST    AND    ANNUITIES.  203 

EXERCISE  XII.  d. 

1.  Find  the  Harmonic  Mean  between  a  and  6. 

2.  If  -4,  G^  JET  denote  the  Arithmetic,  Geometric,  and  Harmonic 
Means  between  two  quantities,  show  that  G  =  y/AH. 

3.  J.,  P,  By  Q  are  four  points  in  line,  and  C  is  half-way  between 
A  and  B.  If  -4P,  AB,  AQ  are  in  H.  P.,  then  OP,  CB,  CQ  are 
in  G.  P. 

4.  If  1,  c,  a  are  in  A.  P.  and  1,  c,  6,  in  G.  P.,  can  c,  a,  6  be 
in  H.  P.  ? 

6.  If  X,  y,  z  be  in  H.P.,  a,  sc,  6  in  A.  P.,  and  a,  z,  b  in  G.P., 
show  that 

-2 


'-(-'>{©'-(!)'}■ 


6.  Three  numbers  are  in  G.  P.  If  the  first  two  be  each  increased 
by  1,  the  result  is  in  A.  P. ;  and  if  2  be  then  added  to  the  third,  the 
result  will  be  in  H.  P.    Find  the  numbers. 


INTEREST    AND    ANNUITIES. 

171.  Let  F  denote  the  principal,  r  the  rate  per  unit,  t 
the  time  in  years,  and  A  the  sum  of  the  principal  and 
interest  at  the  end  of  t  years. 

Then  Pr  =  the  interest  for  1  year. 

(1)  If  the  interest  is  simple,  this  interest  remains  the 
same  for  every  year,  and  in  t  years  becomes  Prt,  And 
adding  the  principal  gives 

which  is  the  relation  connecting  the  quantities  in  simple 
interest. 

(2)  In  a  case  of  compound  interest,  the  amount  at  the 
end  of  the  first  year  becomes  the  principal  for  the  second 


204  INTEREST    AND    ANNUITIES. 

year ;  the  amount  at  the  end  of  the  second  year  becomes 
the  principal  for  the  third  year,  etc. 

Amount  at  end  of  1st  year  =  P(l  +  r). 
Amount  at  end  of  2d  year  =  P(l  +  r)  (1  +  r) 

=  P(l  +  ry. 

Amount  at  end  of  3d  year  =  P(l  +  ry. 
Similarly, 

Amount  at  end  of  t  years    =  P(l  +  ry, 
or  A  =  P(l  +  ry, 

which  expresses  the  relation  connecting  the  quantities 
in  compound  interest. 

172.  The  present  value  of  a  sum  of  money,  payable 
at  some  fixed  future  date,  is  that  sum  which  put  at 
interest  will  amount  to  the  given  sum  at  the  given  date. 

Ex.  1.  What  is  the  present  value  of  a  sum,  S,  payable  t  years 
hence,  money  being  worth  r  per  unit. 

Let  V  be  the  value. 

Then  v(l  +  rV  =  S.     .:   V= — 

^        ^  (1  +  ry 

The  result  is  given  for  compound  interest,  as  in  all 
such  cases  compound  interest  is  the  only  kind  practically 
considered. 

Ex.  2.  If  the  sum  S  pays  yearly  interest  at  rate  r,  and  money 
is  worth  rate  rj,  we  have 

S(l  -\-  ry  =  amount  of  S  in  t  years,  rate  r, 
and  V(l  -\-  r{)*  =  amount  of  Fin  t  years,  rate  r^, 

and  as  these  must  be  equal, 


--iHfJ- 


OF    ANNUITIES.  205 

Ex.  3.  A  loan  of  $  5000  pays  interest  annually  at  4  %  for  4  years, 
and  is  to  be  then  paid  in  full.  What  is  its  present  value,  reckoning 
money  at  6  %  ? 

V=  5000  f  ^-^y=  14633.28 


/L04y^, 

Vi.oeJ 


OF    ANNUITIES. 

173.  An  annuity  is  a  fixed  payment  of  money  made  at 
stated  and  equidistant  intervals. 

If  the  payments  continue  for  a  definite  time,  it  is  an 
annuity  certain,  or  a  fixed  annuity  ;  if  they  continue  only 
during  a  person's  life,  it  is  a  life  annuity ;  and  if  they 
continue  for  all  time,  it  is  a  perpetuity. 

Annuities  may  pay  annually,  or  semi-annually,  or 
quarterly,  or  at  any  other  stated  times ;  but  as  the  prin- 
ciples are  the  same  in  dealing  with  all  of  these,  we  shall, 
unless  otherwise  stated,  consider  the  payments  as  being 
made  annually. 

Problem.    To  find  the  present  value  of  a  fixed  annuity. 

Let  P  be  the  annual  payment,  r  the  rate  per  unit  of 
interest,  t  the  number  of  years  the  annuity  is  to  run, 
V  its  present  value,  and  let  R  stand  for  1  +  r. 

Let  us  suppose  that  the  annuity  is  paid  into  a  bank, 
and  left  there  for  t  years  from  the  time  of  purchase,  to 
accumulate  at  compound  interest. 

The  1st  payment  draws  interest  for  t  —  1  years,  and 
amounts  to  PR~^. 

The  second  payment,  similarly,  amounts  to  PR*^. 

The  3d  payment  amounts  to  PR^~\ 
etc.  etc. 

The  last  payment  is  PR^~^  or  P. 


206  OF   ANNUITIES. 

Therefore  the  whole  amount  is 

K  —  1 

Now  if  the   purchase  money  were  deposited  in  the 
same  way,  it  shouhl,  in  t  years,  amount  to  the  same  sum. 
But  F  dollars  in  t  years  amounts  to  F(l  +  r)* 

7?*  —  1 


Whence  V=  P 


E  -1 
1  -  R-' 


Ex.    What  is  the  present  value  of  $  100  paid  annually  for  6  years 
at  10  %  compound  interest  ? 

Here  72*  =(1.1)6  =  1.77156  ;  and  R-'  =  0.56447. 

Then  F  =  100  x  ^  ~  ^'^^^^^  =  $  435.53. 

0.1 

Cor.    When  t  =  yo,  the  annuity  becomes  a  perpetuity, 
and  for  its  present  value 


174.  When  an  annuity  does  not  begin  to  pay  until 
after  the  lapse  of  a  number  of  years,  it  is  said  to  be 
deferred,  or  to  be  in  reversion. 

Problem.  To  find  the  present  value  of  an  annuity  in 
reversion. 

Let  p  be  the  number  of  years  the  annuity  is  deferred, 
and  let  t  be  the  number  of  years  through  which  its  pay- 
ments run. 


OF   ANNUITIES.  207 

The  amount  of  the  annuity  at  the  end  of  79  +  ^  years  is 


1- 

-R 

? 

and  the  amount  of  V  for  the 

same  time 

is 

VR'^', 

and  these  must  be 

equal. 

R'- 

1 
1~ 

n 

R 

r 

-"{^'l 


Cor.  When  t  =  cc,  we  have  as  the  present  value  of  a 
deferred  perpetuity 

P 

rR^' 

Ex.  A  young  man,  at  the  age  of  19,  will  come  into  a  property 
at  23  that  will  pay  him  $  1000  yearly  during  his  life.  If  his  life 
probability  at  23  is  40  years,  how  much  is  his  annuity  now  worth, 
money  being  at  6  %  ? 

Here        P  =  1000,  p  =  4,  ^  =  40,  r  z^  0.06,  B  =  1.06. 
1000  f       1  1 


And         V  = 


{-1 


0.06   C  (1.06)4      (i.06)-t4. 

This  cannot  be  conveniently  Avorked  out  without  the  use  of 
either  logarithms,  or  tables  of  the  powers  of  1.06. 
The  value  is  $11918,  to  the  nearest  dollar. 

175.  An  annuity  which  has  not  been  paid  for  a  number 
of  years  is  said  to  be  foreborne.  The  present  value  of  a 
foreborne  annuity  is  the  cash  value  of  all  due,  together 
with  the  present  value  of  the  annuity  as  continued  into 
the  future. 

To  find  the  present  value  of  a  foreborne  annuity.  Let 
the  annuity  be  foreborne  for  q  years. 

Its  cash  value  is     P  •  ^  ~  ^'  or  P  •  ^^-^-. 
1-R  r 


208  OF   ANNUITIES. 

1  —  R~^ 

And  its  value  for  the  future  is    P ,  t  being 

r 

the  number  of  years  it  is  to  continue. 

...  v=p[^^]. 

Cor.  If  t=cc,  we  have  for  the  present  value  of  a 
fore  borne  perpetuity 

r 

176.    The  following  problem  is  of  special  importance. 

Problem.  A  corporation  borrows  A  dollars,  which  is 
to  be  paid  in  t  equal  annual  instalments,  each  instalment 
to  cover  all  interest  due  at  the  time  of  payment.  To 
find  the  value  of  each  instalment. 

A  part  of  the  instalment  goes  to  pay  interest,  and  the 
remainder  goes  to  reduce  the  debt. 

Let  a,  b,  c,  •  •  •  t,  be  the  parts  applied  in  successive  years 
to  the  reduction  of  the  debt,  and  let  p  be  one  of  the 
annual  instalments. 

Then, 
1st  payment    =p  =  a  -{-  Ar. 
Eeduced  debt  =  A  —  a. 

2d  payment     =p  =  b  +  (A  —a)r  ;  whence  b  =  aB. 
Eeduced  debt  =  A  —  a—  b  =  A  —  a  —  aR. 
3d  payment     =2)  =  c-\-{A  —  a  —  aR)r ;  whence c  =  aR^, 
Eeduced  debt  =  A  — a  —  b  —  c  =  A  —  a  —  aR  —  aR\ 

tth  payment     =  2)=t-^\A—a  —  aR—aR^ aR-^lr, 

Eeduced  debt  =  A  —  \a -i- aR -\- aR^ -\ aR-^}, 

But  after  the  tth  payment  the  debt  must  be  nothing. 


,.-ix    ^B'-l 


OF   ANNUITIES.  209 

.'.  A  =  a(l  +  E  +  B'  +  "^E'-')  =  a 

But  a=p  —  At,  and  eliminating  a  between  these  gives 

p  =  A' -> 

^  R-1 

which  is  the  value  of  the  annual  payment. 

EXERCISE   XII.   e. 

1.  A  mortgage  for  ^1200  pays  $400  annually  for  3  years  with- 
out interest.  What  is  its  cash  value  when  drawn,  money  being 
reckoned  at  6  %  compound  interest  ? 

2.  Find  the  present  value  of  the  mortgage,  of  Ex.  1,  if  it  pays 
interest  at  4  %,  while  money  is  worth  5  %. 

3.  An  annual  annuity  of  $1000  is  foreborne  for  6  years,  and 
is  to  run  8  years  in  all.  What  is  it  now  worth,  money  being 
reckoned  at  4  %  interest  ? 

4.  A  man  borrows  $  500  on  a  mortgage  and  wishes  to  pay  prin- 
cipal and  interest  in  5  equal  instalments.  What  is  the  amount  of 
each  instalment,  calculating  interest  at  6  %  ? 

5.  A  corporation  borrows  $30000  at  4%  interest,  and  is  to  repay 
principal  and  interest  in  30  equal  instalments.  What  is  the  value 
of  an  instalment  ? 


CHAPTER  XIII. 

Permutations,   Combinations,   Binomial 
Theorem. 

177.  If  from  n  different  objects  we  form  groups  each 
containing  r  objects,  such  that  no  two  groups  contain 
the  same  assemblage  of  objects,  each  group  is  called  a 
Combination,  and  the  possible  number  of  such  groups  is 
the  number  of  combinations  of  n  things  r  together. 

This  number  is  symbolized  by  "C^,  and  read  'n  objects 
combined  by  ?''s.' 

Thus  ^Cg,  taking  letters  as  objects,  is  4,  and  the  several 
groups  or  combinations  are 

ahc,  ahd,  acd,  and  bed. 

Similarly  ^Og  has  for  its  groups  a5,  ac,  ad,  ae,  he,  bd, 
be,  cd,  ce,  and  de ;  or  10  in  all. 

The  combination  abc  is  the  same  as  acb,  the  same  as 
bac,  etc.,  since  all  have  the  same  assemblage  of  letters. 

If,  however,  we  take  relative  position  into  considera- 
tion, abc  and  acb  are  not  the  same,  since,  although  they 
contain  the  same  letters,  the  letters  are  differently  ar- 
ranged. Distinguishing  different  groups  in  this  way, 
each  group  is  called  a  Permutation,  and  the  possible 
number  of  such  groups  is  the  number  of  permutations 
of  n  things  r  together. 

This  number  is  symbolized  by  "P^. 
210 


PERMUTATIONS.  211 

The  combination  ahc  gives  6  permutations :  dbc,  acb, 
bac,  bca,  cab,  and  cba;  and  as  each  combination  may  be 
treated  similarly,  the  number  of  permutations  of  4  letters 
3  together  is  24 ;  or  ^F^  =  24. 

PERMUTATIONS. 

178.  Problem.  To  find  the  number  of  permutations 
of  71  things  r  together,  n  being  greater  than  r. 

If  we  have  r  boxes,  A,  B,  C,  •••,  etc.,  into  each  of  which 
one  of  the  n  letters,  a,b,c*"  is  to  be  put,  the  number 
of  ways  in  which  the  distribution  can  be  effected  is  the 
number  of  permutations  of  n  things  r  together. 

In  filling  box  A,  we  may  choose  any  one  of  the  n 
letters,  and  we  have  therefore  n  choices. 

Having  filled  A,  we  have  n  —  1  choices  in  filling  B, 
and  any  one  of  these  n  —  1  choices  may  be  combined 
with  the  n  choices  in  filling  A, 

Hence  in  filling  A  and  B  we  have  n(n  —  1)  choices. 

Similarly,  infilling^,  B,  and  (7,  we  have  n(n  —  1)  (n  —  2) 
choices,  and  so  on  through  the  r  boxes. 

.-.  "P^  =  n{n  -  1)  (n  -  2)  ...  {n-r-\- 1). 

179.  Multiply  the  value  found  for  "P^  by  the  unit 
fraction 


(it-r)(yi-r-l)...3.2.1 
1.2.3...  (n-r-l)(n-r)' 


and  we  obtain 


,p^l.2.3...(n-2)(n-l)n 
1.2.3...  (n-r) 


212  l>ERMtJTATlOKS. 

The  continued  product  of  m  consecutive  natural 
numbers,  beginning  at  1,  is  called  factorial  m,  and  is 
symbolized  as  m  !,  or  [m. 

rt  f 

Hence  "P.  =  - 


{n  —  r)\ 
Cor,   When  r  =  n,  we  have 

"P„  =  71 !  (Art.  178)  =  "^  (Art.  179), 

and  hence  0 !  must  be  interpreted  as  meaning  1. 

Ex.  1.   The  number  of  permutations  of  12  things  5  together  is 
i2p^  =  12  .  11 .  10  .  9 .  8  =  95040. 

Ex.  2.  The  number  of  permutations  of  n  things  3  together  is 
14|  times  the  number  of  permutations  of  n  —  2  things  2  together, 
to  find  n. 

Here 


n{n-V){n-2)  _U_n 


(7i-2)(7i-3)        3       n-3 

Whence  n  =  12,  or  3f . 

As  n  must  be  integral,  its  value  is  12,  and  3|  must  be  rejected 
as  being  inapplicable  to  the  nature  of  the  problem. 

Nevertheless,       (3f  •  2f  •  If)  -  (If  •  Of)  =  ~%K 

180.  In  the  permutations  "P^  to  find  how  many  con- 
tain a  particular  object  or  letter,  as  a. 

Putting  a  aside,  we  form  a  group  of  r  —  1  from  n  —  1 
objects,  and  this  can  be  done  in 

(?i  —  1)  (n  —  2)  •  •  •  (n  —  r  +  1)  ways. 

In  each  of  these  groups  a  can  have  r  positions;  namely, 
from  preceding  all  the  other  letters  to  following  them  all. 


PERMUTATIONS.  213 

Hence  the  number  of  permutations  containing  a  is 
r(n  — l)(n  — 2)  ...  (n  —  r  +  l). 

Similarly,  the  number  of  permutations  containing  two 
particular  letters,  as  a  and  6,  is 

r(r-l)(n-2)  ...(n-r  +  1). 

Containing  3  particular  letters  together,  it  is 

r(r  -  1)  (r  -  2)  (n  -  3)  ...  (n  -  r  + 1). 

etc.  etc. 

Ex.  How  many  numbers  can  be  made  from  5  figures,  1,  2,  3,  4,  5, 
three  at  a  time ;  and  how  many  of  these  will  contain  1  ?  How 
many  contain  1  and  2  ? 

1st.  «P^  =  5  .  4  . 3  =  60. 

2d.    r(w-l)(n-2)=3.4.3  =  36. 

3d.    r(r- l)(n-2)=3.2.3  =  18. 

181.  To  find  the  number  of  permutations  of  n  things, 
all  together,  when  u  of  the  things  are  alike. 

Denote  the  number  by  ''F(u). 

If  the  u  things  were  all  different,  they  would  in  them- 
selves give  rise  to  u !  permutations,  each  of  which  com- 
bined with  each  of  the  permutations  of  ''P(u)  would  give 

Or  "P(w)  =  — . 

ul 

Similarly,  if  **P(u)  (v)  denotes  the  number  of  permu- 
tations all  together,  when  u  articles  are  alike  of  one 
kind,  and  v  articles  are  alike  of  another  kind, 


214  PERMUTATIONS. 

Ex.  How  many  permutations  can  be  made  from  the  letters  in 
Mississippi  9 

Here  there  are  11  letters,  of  which  4  are  i's,  4  are  s's,  and  2 
arep's. 

.-.  ^P(u)  (v)  (w)  =  —111—  =  34660. 

If  the  permutations  were  to  be  such  as  not  to  have 
repeated  letters,  we  have  only  4  different  letters,  and 
the  number  is 

«P„  =  4  .  3  .  2  . 1  =  24. 


EXERCISE  XIII.  a. 

1.  Find  the  values  of  — 

2.  Given  «P^  =  S^-iP^,  to  find  "Pg. 

3.  Given  "Pg  =  V  ""^P^,  to  find  n. 

4.  How  many  permutations  can  be  made  from  the  letters  in 
College  ?  in  Oporto  f  in  Amsterdam  ? 

In  each  of  these  how  many  permutations  would  have  letters 
repeated  ? 

5.  A  person  writes  at  random  3  of  the  figures  1,  2,  3,  4,  5,  6. 
What  is  the  probability  that  they  will  be  consecutive  and  in 
ascending  order  ? 

6.  With  four  different  consonants  and  a  vowel,  how  many  words 
of  3  letters  can  be  made  having  a  vowel  in  each. 

7.  The  figures  from  1  to  9  are  written  down,  and  a  person  erases 
3  figures  at  random.  What  is  the  chance  that  the  figures  erased 
may  be  consecutive  ? 

8.  Six  points  are  taken  on  a  circle.  In  how  many  different 
ways  may  they  be  joined  by  twos  ? 


COMBINATIONS.  215 


COMBINATIONS. 

182.  As  a  combination  has  no  reference  to  arrange- 
ment, each  combination  of  r  articles  can  give  rise  to  r ! 
permutations  r  together. 

Hence  "P,  =  r !  x  "(7^. 

r !      r !  (n  —  r)  ! 

or,  by  reduction,     "0^  =  ^  •  ^"7     •  ^^^ to  r  factors. 

Cor.  Since  "(7^  is  necessarily  an  integer,  it  follows 
that  the  continued  product  of  any  r  consecutive  integers 
is  divisible  by  factorial  r. 

183.  Substituting  n  —  r  for  r  gives 

{71  —  r)  !  r ! 

Or  the  number  of  combinations  of  n  things  r  together 
is  the  same  as  the  number  n  —  r  together. 

This  is  quite  self-evident,  for  every  time  we  take  an 
assemblage  of  r  things  out  of  n  things  we  leave  an 
assemblage  of  n  —  r  things,  and  the  numbers  of  the 
two  assemblages  must  necessarily  be  equal. 

"Or  and  "C7„_r  ^^re  complementary  combinations. 

184.  Since  "C^=  ^.^llll.^JIl^...,    the  number  of 

combinations  will  increase   with   r  as  long  as  the  last 
fractional  factor  is  greater  than  1.     But  this  factor  is 


216  COMBINATIONS. 

^"~^"^    ;  and  while  this  is    >1,  the  number  of  com- 
r 

binations  increases. 

71  4- 1 
.-.  71  — r  +  l>r,  or  t<--^ — 

And  T  is  to  be  the  integer  nearest  to  but  less  than 
1(71  +  1).     Therefore 

If  71  is  even,  the  value  of  r  which  makes  "(7,.  greatest  is 
r  =  ^7i;  and  if  n  is  odd,  the  value  of  r  is  ^(ti  — 1),  or 
1(71  +  1);  the  latter  value  giving  a  unit-factor. 

Ex.   r2a^^l2-ll-10>9^12.11.10.9.8.7.6.5^,^^^^^^^ 
*         1.2.3.4  1. 2. 3. 4. 5. G. 7.0  ^ 

12.11.10.9.8.7^^,^ 
:  ^        1.2.3.4.5.6 

n  /^  ^  11 '  10  '  9  .  8  .  7  ^  11 .  10  .  9  .  8  ♦  7  .  6  ^  -n  ^ 
^        1.2.3.4.5  1.2.3.4.5.6  ^* 

185.  In  the  combinations  "(7^,  to  find  the  number  of 
times  any  particular  object,  as  a,  will  be  present. 

If  we  form  "~^(7r_i  from  all  the  letters  except  a,  taken 
r  —  1  together,  we  can  place  a  with  each  of  these  groups, 
and  we  then  have  all  the  combinations  of  n  letters   r 
together  containing  a. 
^    Thus  a  occurs  "~^(7^_i  times. 

Similarly,  ah  occurs  "^(7,._2  times,  etc. 

Ex.   Out  of  a- guard  of  12  men  5  are  drafted  for  duty  each  night. 
Relatively,  how  often  will  A  be  on  duty  ?     How  often  will  A  and  B 
be  together  on  duty  ?     How  often  will  A  be  on  duty  without  B  ? 
.     As  ^2^5  =  792,  this  is  the  total  number  of  different  drafts., 

1.  A  is  on  duty  "C^  -  330  times  out  of  792. 

2.  A  and  B  are'together  ^^C^  =  120  times  out  of  792. 

3.  .'.  A  is  present  without  B,  210  times  out  of  792. 


i 


BINOMIAL  THEOREM.  217 

EXERCISE  XIII.  b. 

1.  If  a  =  **(73,  and  b  =  "Pj,  find  the  relation  between  a  and  b. 

2.  If  a  =  "Or,  and  b  —  "-^Pr-i,  find  the  relation  between  a 
and  b. 

3.  At  an  election  there  are  10  candidates,  of  which  4  are  to 
be  elected.  If  a  man  may  vote  for  1  or  more  up  to  4,  how  many 
different  votes  can  he  cast  ? 

4.  If  "C„_i  =  "»+i(7„»-i,  then  m(m  +  1)=  2n. 

5.  Prove  that  "O-  +  "CV-i  =  «+iCr. 

6.  Prove  that  w{«0r  +  «-iCr-i}=(w  +  r)«(7r. 

7.  3  black  and  2  white  balls  are  put  into  a  bag ;  what  is  the 
chance  of  drawing  2  black  balls  at  a  single  drawing  of  2  balls  ? 

THE    BINOMIAL    THEOREM. 

186.  The  expansion  of  (l+o;)**,  with  n  a  positive 
integer,  is  the  simplest  form  of  the  binomial  theorem,  or 
binomial  formula.  The  theorem  is  then  generalized  and 
adapted  to  any  value  of  n  whatever.  The  simplest  case 
is  first  established. 

I.     w  A  Positive  Integer. 

The  number  of  terms  in  2a  with  n  letters  is  "Ci;  the 
number  of  terms  in  2a6  is  "C2 ;  and  generally  the  number 
in  Sa6---r  is  "(7^. 

These  statements  are  self-evident. 

Now   (x  -]-  a)  {x  '\-  b)  (x  -{-  c)  "-  to  n  factors  is 


218  BINOMIAL  THEOREM. 

And  making  a  =  6  =  c  =  -..==l,  we  obtain 

(x  + 1)"  =  ic"  -f  "Cio;"-^  +  "C2a;"-2  +  •••  +  "C„. 

Or,  since     "(7„  =  1,  "(7„.i  ="(7i,  "(7„_2  =  "Oa,  etc. 

(1  +  0?)"  =  1  +  "Cjo;  +  ""C^  +  -  "CX  +  -     (A) 

Also  writing  the  factor  values  of  "Ci,  "Cg,  etc., 


1  •  ^ 


(^)  and  (B)  are  common  forms  of  the  binomial  the- 
orem; and 

iC, 

r! 

the  (r4-l)th  term   from  the  beginning,  is  called  the 
general  term. 

Knowing  the  particular  value  of  each  coefficient,  these 
coefficients  are  often  denoted  by  a  single  letter  with 
subscript  numbers,  and  the  theorem  then  becomes 

where  Cq  =  1,  Ci  =  n,  Cg  =  ^^^"~  ^,  etc. 

Ex.  1.     (a  +  «)«  =  a^(l  +  -V  =  a^(l  +  Cj  -  +  c.,^  +  r-\ 

=  a«  +  Cia«-ix  +  Caa"-2x2  +  •••  Cra«-*'«^  H 

and  thus  the  expansion  of  any  binomial  depends  upon  that  of 
(1  +  X). 

Ex.  2.     (1  -  iK)«  =  1  +  Ci(-  cc)  +  C2(-  a;)2  -f  CgC-  a)8  +  ... 
the  signs  being  alternately  +  and  — . 


BINOMIAL  THEOREM.  219 

187.    Since  (l  +  x^  =  l-\-''CiX-i-"C^'^ -{ h"0„a;^ 

making  ic  =  1  gives 

2»=l4-"Ci  +  "(72  +  "03+...+"a. 
.-.  The  total  number  of  combinations  of  n  things  taken 
1  at  a  time,  2  at  a  time,  and  so  on  to  n  at  a  time,  is  2"  —  1, 
Also,  the  sum  of  the  coefficients  of  the  expansion  of 
(l  +  a;)»is2". 

Also,  since  (1— a;)"=l—  CiX  +  CzX^  —  c^pc^  -\ ,  by  mak- 
ing a;  =  1  we  have 

0  =  1  4-  Cg  +  C4  H (ci  +  Cg  +  C5  +  •••)• 

Or,  the  sum  of  the  odd  coefficients  in  the  expansion  of 
(l-{-xY  is  equal  to  the  sum  of  the  even  coefficients. 
Ex.  To  find  the  sum  of  the  squares  of  the  coefficients  of  (1  +x)". 
(1  +  x)«  =  Cq+  CjX  +  C2X2  ^  (.3^3  +  ...  +  c„x» ; 
(x  +  1)«  =  CqX^  4-  c^x'^-^  +  C2a;«-2  ^  C3X"-3  -\-  ,..Cn. 
The  coefficient  of  ic"  from  the  product  of  the  right-hand  mem- 
bers is 

Co2+Ci2  +  C22+...C„2. 

But  the  coefficient  from  the  product  of  the  left-hand  members  is 
the  coefficient  of  x»  from  the  expansion  of  (1  +  x)2« ; 

that  is,  2«(7^  or  i?^. 

n!  w! 

..  Co  -hCi   +C2  +  —  C„  -  ^^^^2 

Cor.  1.  This  last  expression  on  the  right  is  the  number  of  per- 
mutations of  2  n  articles,  when  half  of  them  are  alike  of  one  kind, 
and  the  other  half  alike  of  another  kind. 

Cor.  2.    The  coefficient  of  x"-2  from  the  right  product  is 


And  from  the  left  it  is 


(^ny. 


(n-2)!(w  +  2)! 
The  student  is  left  to  generalize  this. 


220  BINOMIAL   THEOREM. 


EXERCISE  XIII.  c. 

1.  Write  the  general,  or  (r  -f  l)th,  term  in  the  expansions  — 

i.    (a  +  x)".  ii.    (a  -  x)«.  iii.    (1  +  x)2». 

2.  Show  that 

(a  +  x)»  _  a»    x^  ,      g^-^      ^  x^  ,      a^-2      .  ?^  _!_...  4.  «°    ^. 
n!  w!  *0!      (ri-1)!  *  H      (n-2)!  *  2!  0!  *  n\    . 

3.  Find  the  10th  term  in  the  expansion  of  (1  +  xy^. 

4.  What  is  the  factor  which  changes  the  (r  +  l)th  term  into  the 
(r  +  2)th  term  ? 

5.  Find  the  value  of  r  that  the  factor  of  question  4  may  be  the 
last  one  greater  than  unity. 

6.  Find  the  greatest  coefficient  in  the  expansion  of  (2  4-  3  x)^. 

7.  Show  that  the  6th  term  has  the  greatest  coefficient  in  the 
expansion  of  (3  +  2x)i2. 

8.  Prove  that  "Cr  +  "Cr-i  =  »+^Cr. 

9.  In  the  expansion  of  [  x  +  -  ]    show  that  the  coefficient  of  x**  is 


Qn-r)liin  +  r)\ 
II.     n  A  Negative  Integer. 

188.   Let  the  expression  "("  +  l)('^  +  ^)-(^^  +  ^-^) 
be  denoted  by  ""H^. 

Then  "H,  +  »+'^,_,  =  n(n  +  l)...{n  +  r-l) 

r ! 

(n+1)  (w+2)  -(n+r-1)  ^(n  +  l)(n  +  2)-(n  +  r) 
(r-iy.  r! 

=  »+'ir, (D) 


BINOMIAL   THEOREM.  221 

Now  we  know  from  division  that 

{1-xy 

^1    ^1.2     ^1.2.3  {n-iy. 

=  l  +  'H,x  +  'Hox^  +  'B^^  +  ...  ^jy^.io:"-^  +  ... 
Therefore  let  us  assume  that 
1 


(I-:.-)" 


=  l+"iyia?+"iy2^+"^3^'+  •••+"Jy.i»'+  -.   (^) 


Divide  both  sides  by  1  —  a; ;  the  left-hand  member  be- 
comes   r,  and  the  right  as  follows  : 

(1  -  xy+^ 


because  '^^^  +  "+^^,_i  =  "+W„  from  (D) . 

.-.        __^^  ^^^  =  1  +  "-^^J^io?  +  -+^^2^^  +  "^^^3^^ 

Hence  if  the  expansion  is  true  for  any  value  of  n,  it  is 
true  for  the  next  greater  value.  But  it  is  true  for  n  =  2, 
and  therefore  for  n  =  3,  for  n  =  4:,  etc. :  that  is,  it  is 
generally  true.     And  the  expansion,  (E),  is  true. 

XT        /I        \n     -I            I  n(n—l)     n(n—l)(n^2)   , 
Now  {l—xy=l—nx+    ^         ^ ^^ — ToQ '^ 


222  BINOMIAL   THEOREM. 

Change  n  to  —n,  and  this  becomes 

^       ^  1.2  1.2.3 

=  1  +  ".^laJ  +  "^^2^^  +  "^3^^  +  •••,  agreeing  with  (jEJ). 

And  we  see  that  the  general  form  of  the  binomial 
theorem  holds  good  for  all  integral  values  of  n,  positive 
or  negative. 

With  n  positive,  the  series  terminates  when  ^~^  "^  ■ 

r ! 
becomes  zero ;   i.e.  when  r  =  n  +  1.      Or  the  series  con- 
tains n  + 1  terms. 

With  n  negative,  however,  the  series  is  infinite,  as 
w(n4-l)(n4-2)...  cannot  become  zero  by  extending  the 
number  of  factors. 

189.   To  interpret  "5;. 
1 


1 

1-hx 

1 


1  —  ex 


=  1  +  5a;  4.  52^  +  53^  _f.  ...  +  ^r^r  ^  .. 

=  1  4-  ex  +  c^x^  4-  cV  H h  e^'x*'  +  ... 


By  multiplying  n  such  equations  together,  the  coeffi- 
cient of  X*",  on  the  right,  is  the  sum  of  all  the  homogeneous 
terms  of  r  dimensions  that  can  be  made  out  of  n  letters. 
But  if  we  make  a  =  &  =  c=...  =  l,  the  left-hand  product 
becomes  (1  —  x)  ~",  and  the  coefficient  of  x""  is  ""11^. 


BINOMIAL   THEOREM.  223 

.•.  "jET^  =  the  number  of  homogeneous  terms  of  r  dimen- 
sions which  can  be  formed  from  n  letters,  and  their 
powers. 

Thus,  if  n  =  4  and  r  =  2,  ^Hr  =  ^H^  =  j^  =  10  ;  that  is,  there 

are  10  homogeneous  terms ;  namely,  a^,  6"^,  c-^,  c2^  ab,  ac,  ad,  be,  bd, 
and  cd. 

It  is  well  to  notice  that  if  we  denote  ""Hi  by  /ij,  "^2  by 
^2j  6tc., 

(1  +  xy   =  1  -h  Cio;  4-  CgO^  +  CgiB^  H signs  all  +. 

(1  —  a;)~"  =  1  +  hiX  +  ^2^  +  /igO^  H signs  all  +. 

(1  —  xy   =  1  —  CiX  4-  Cgo;^  —  CgO^  H signs  alternate. 

(1  4-  a?)  ~"  =  1  —  /ii^;  +  ^2^  —  h^a^  -\ —  •  •  •  signs  alternate. 

EXERCISE  XIII.    d. 

1.  Expand  (1  —  jc)-*,  and  show  that  the  coefficients  are  the 
sums  of  the  coefficients  of  (1  —  x)-\ 

2.  Expand   ^t_^ —  in  ascending  powers  of  x,  and  find  the 

1  +  x-\-  x'^ 
coefficient  of  x"  in  the  expansion. 

(1  4.  x\*» 
l-xj 

4.  Prove  that  (l+x)«=2»  \  l-/^^  .lli-?+ ^  Yln^V-^- ...  \. 

(  1+x       \i-tx/  y 

5.  Find  the  coefficient  of  x^^^  in    ^-^^  , 

(l-x)2 

6.  Find  the  coefficient  of  x"  in  the  expansion  of 

(1  -2x4-3x2-4x3  + )-«. 

7.  Show  that  if  n  is  a  positive  integer,  (5  +  2^Qy  is  odd  in  its 
integral  part. 


224  BINOMIAL   THEOREM. 

Since  52  -(2^6)2  =  1  and  5  +  2^6  >  1,  .-.  6  -  2^6  <  1,  and 
is  accordingly  a  proper  fraction. 

.  •.  5«  -  Ci  5"-i .  2  V^  +  Cj  6"-2  22 . 6  -  +  ...=/'  =  proper  fraction. 
5«  +  Ci  5^-1 . 2  V6  +  C2  5«-2  22 .  6  ++  —  =  /+/  =  an  integral 
part  +  a  proper  fraction. 

. •.  2{5«  +  C2 •  5«-222 . 6  +  ...}  =  /  +  /  +  /=  an  integer. 

.  *.  f  +  f  must  =  1 ;  and  as  /  +  /  +  /  must  be  even,  /must  be  odd. 

8.  Show  that  the  coefficient  of  x'^  in  the  expansion  of  (1  +  xy^ 
is  double  the  coefficient  of  x"  in  the  expansion  of  (1  +  x)2«-i. 

9.  By  the  Binomial  theorem  find  99^. 

10.  Prove  that  c^  -  2  Cj  +  3  Cj  -  +  •••  +  (-  l)^(w  +  l)c„  =  0. 

11.  Show  that  »/r^  =  «+^-iCV. 


III.     n  A  Fraction. 

190.  With  n  fractional  there  are  certain  difficulties  in 
the  Binomial  theorem  which  we  cannot  here  explain ; 
and  no  very  satisfactory  proof  of  the  theorem  with 
n  fractional  can  be  given  without  involving  higher  con- 
siderations than  occur  in  this  w^ork. 

Several  methods,  however,  will  furnish  proofs  which 
are  morally  sufficient. 

The  following  is  Euler's. 

(1  +  a;)''  is  a  function  of  n ;  denote  it  hj  fn. 
Then  (l-\-x)"'=fm,Vind  (l+a^)"*+"=/(m+n). 

But  (1  +  x)"^ '  (1  -f  xY  =  (1  +  .1')'"^"  by  the  index  law. 

.-.  fm'f)i=f{m-\-n). 
Similarly,      fm  -fn  -fp  =f{m  +  71  -\-p)  j 
and  generally, 

fm  'fn  'fp'"k  factors  =f{m  +  n  -i-p'-'Jc  terms). 


BINOMIAL   THEOREM.  225 

h 


Now  let  m  =  n=p=  '*'  — 

k 

and  f(^={fh)\  =  {l+x)i. 

But,  fn  =  l  +  nx  +  ^^^^^^x'  + 

1  •  z 


h 


i-^). 


Or  (l  +  a;)Ll+|a;4-^^^^a^  + 

And  the  form  of  the  Binomial  theorem  holds  good  for 
n  fractional. 


-1 


Cor.    If  we  make  k  =  —  1,  f(—h)  =  (fh) 
But 

and  (//()-'=  ^  ^ 


fh      (l+x)" 
1 


.  =  (!  +  «')• 


-ft 


■•   {l  +  x) 

which  proves  the  theorem  for  negative  indices 


226  BINOMIAL   THEOREM. 

EXERCISE  XIII.  e. 

1.  To  find  VT^^,    This  is  (1  -  x)*, 

and  (1  -  a;)i  =  1  -  Jx  +  Ki^niix^  -  i(L—0(isz^y.9  ^  ... 
^  ^  ^1-2  1-2.3  ^ 

=  1 -_  1  a;  -  J_x2  - -1^x3  - -i-lil^x* -- ... 
^        2.4  2.4.6         2.4.6.8 

2.  Write  the  general  term,  (r  +  l)th,  of  Ex.  1. 

3.  Expand  (I  -\-  x)^  in  ascending  powers  of  x. 

4.  Find  the  approximate  value  of  a(l  —  x)  Vl  +  x^  when  sc^ 
is  so  small  as  to  be  rejected. 

5.  Expand  (14-  x)%  and  find  the  result  when  x  =  0. 

6.  Expand  (  1  +  -  )  ,  and  find  the  result  when  x  =  oo. 


(-i)- 


7.  Find  the  value  of  $  1  compounded  every  moment  for  t  years 
at  r  %  per  annum. 

1 

8.  By  expanding  (1  +  x)"  and  making  x  =  3  and  n  =  2,  show 
that  2  is  the  limit  of  the  series 


2      1.2V2/       1.2.3V2>/ 


9.   Find  the  limit  of  the  series  to  infinity  — 

l  +  }-2-J.22  +  /^.23-^VV.244-- 


10.   If  e  =  ( 1  +  -  )  where  x  =  oo,  show 


that 


^21     3!  n\ 


CHAPTER  XIV. 
Inequalities. 

191.  When  two  unequal  expressions  are  compared, 
particularly  with  the  purpose  of  showing  that  the  ex- 
pressions are  not  equal,  the  whole  is  called  a  non-equatlop^ 
or  inequality. 

An  inequality  employs  the  signs  >  and  <  between 
its  members,  and  sometimes  the  signs  ^fc,  read  not  equal 
to,  >,  read  not  greater  than,  and  <,  read  not  less  than. 

It  usually  happens  that  some  values  of  the  variables 
will  change  an  inequality  to  an  equality,  i.e.  an  identity. 

The  working  rules  for  inequalities  being  in  some 
respects  different  from  those  for  equations,  must  be  here 
established. 

I.    Let  CL>b, 

and  let   all   the   quantitative   symbols   denote   positive 
quantities. 

1.  Let  a  =  b  +  /3,  and  add  p  to  both  sides. 
.-.  a+p  =  b-^p-\-p,  ov  a-{-p>b+p. 

2.  Subtract  p  from  both  sides,  and 

a  —p  =  b—p-\- 13,  OT  a  —p  >b—p. 

Hence  the  same  quantity  may  be  added  to  or  subtracted 
from  both  members  of  an  inequality ;  and  hence  a  term 
may  be  transposed  from  one  member  to  the  other  by 

227 


228  INEQUALITIES. 

changing  the  sign  of  the  transposed  term,  without  affect- 
ing the  character  of  the  inequality. 

3.  Subtract  both  members  from  p. 

Then  p  —  a=p  —  h  —  P]  ov  p  —  a<p  —  b,  and  the 
character  of  the  inequality  is  changed. 

Therefore,  if   both  members  be  subtracted  from  the   " 
same  quantity,  the  character  of  the  inequality  is  reversed. 

4.  ma  =  mb  +  w/5 ;  or  ma  >  mb. 

m       m     m         mm 

Hence,  if  both  members  be  multiplied  or  be  divided 
by  the  same  quantity,  the  character  of  the  inequality  is 
unchanged. 

/>     m         mm  mB        ^„  '^  ^  "^ 

a      b  +  p      b      b{b  +  py         a      b 

Hence,  dividing  the  same  quantity  by  both  members 
changes  the  character  of  the  inequality. 

7.  To  multiply  or  divide  both  sides  by  a  negative 
quantity  is  equivalent  to  exchanging  the  members,  and 
therefore  it  reverses  the  character  of  the  inequality. 

II.   Let  a>b  and  c>d. 

Put      a  =  6  +  A  andc  =  d  +  S. 

8.  a-{-c  =  b-{-d  +  (3-\-S;  ora  +  c>6  +  d. 

9.  a  —  c  =  &  —  d  +  /5  —  S;  from  which  we  cannot  infer 
whether  a  —  c>b  —  dov  <6  —  d. 

If  /5  >  S,  a  -  c>  &  -  d;  but  if  ^  <  S,  a  —  c  <  6  —  d 


IKEQUALITIES.  229 

Hence,  inequalities  of  the  same  character  may  have 
corresponding  members  added;  but  they  do  not  in 
general  admit  of  being  subtracted. 

192.  Inequalities  are  usually  referred  to  certain  stan- 
dard forms,  or  determined  by  fixed  relations. 

(1)   Eor  all  values  of  x  and  y^  except  equality, 

a^  +  1/^  >  2  a^i/. 

Proof.     (x  —  yY  is  essentially  positive,  and  .-.  >0. 
...  r^^f_2xy>0, 

and  x^  +  y^':>2xy (A) 

Ex.  1,  The  sum  of  a  number  and  its  reciprocal  is  greater  than  2. 

x  +  -  >2, 

X 

if  a;2+i2^2x.l. 

And  this  latter  is  true  {A).     /.  the  former  is. 

Ex.  2.  To  show  that  1 -{- a^ -\- a'^>  ^(a  +  a^). 
If  a  is  negative,  this  is  evidently  true,  since  the  left-hand  mem- 
ber is  essentially  positive. 
Let  a  be  positive. 
To  prove  that  2  +  2a2  +  2a*  >  Sa  +  Sa^. 

(a  —  1) (a^  —  1)  =  a*  —  a  —  a^  +  1 ;  and  is  +  when  a  is  +. 

.-.1  -\-a^>a  +  aS. 

Also  1  +a2->2a .    .    .    (Ex.1) 

and  a^  +  a^>2  aK 

.-.  Adding,  2  +  2a'^  +  2a^>Sa  +  Sa^. 

193.  (2)  (x~  — 2/")  (^•"'  —  2/"")  >  0,  if  m  and  n  are  both 
odd  or  both  even  positive  integers. 


230  INEQUALITIES. 

Proof.  If  X  and  y  have  the  same  sign  or  opposite 
signs,  both  factors  have  the  same  sign,  and  the  product 
is  positive. 

Ex.  1.  sfi  +  ^^  x^y  +  xy^  according  as 
x^  —  x^y  +  i/^  —  xy^  -^  0, 

as  (a^  -  2^)  (aJ  —  2/)  ^  0. 

But  (x5-2/5)(x-?/)>0. 

.  •.  x^  +  t/^  >  x^y  -\-  xyK 


EXERCISE    XIV 

1.  If  a,  ft,  c  •••  a  be  any  unequal  quantities  forming  a  cycle,  show 
that  Sa2  >  2a6. 

2.  Showthata2  +  3  52^2  6(a  +  &). 

3.  Show  that  a%  +  ah^>2  a^h'^. 

4.  Show  that  {aP'  +  h^)  (a*  +  h^)  >  {a^  +  &^)2. 

5.  With  three  letters,  ^a%  >  6  ahc. 

6.  L±^  +  ^i«-f  «±^>6,  unlessa  =  6  =  c. 

ahc 

7.  Which  is  the  greater  — 

1.    (a2  +  62)  (c2  4-  ^2)  or  (ac  +  6(^)2  ? 
ii.   m2  +  m  or  m^  +  1  ? 

8.  If  X  is  real,  x2  -  8x  +  22  <  6. 

9.  Under  what  circumstances  isx  +  ->or<4? 

X 

10.   An  isosceles  triangle   is   greater  in   area  than   a  scalene 
tri.mgle  with  the  same  perimeter  and  base. 


CHAPTER    XV. 

Undetermined  Coefficients  and  their 
"Applications. 

194.  Theorem.  If  an  integral  function  of  aj  of  n 
dimensions  is  satisfied  by  more  than  n  different  quanti- 
ties, it  is  satisfied  by  all  quantities,  or  its  coefficients  are 
severally  zero. 

Let        fx  =  ax""  +  6aj"-^  +  cx""'^  -\ sx  +  t  =  0 

be  satisfied  by  the  n  values,  a,  (3,  y**  t. 

Then     /a;  =  a(a;  —  a)  (x  —  /?)  (a;  —  y) ...  (a?  —  t)  =  0. 

Now,  if  it  is  satisfied  by  an  (n  -\-  l)th  value  z, 

fz  =  a(z  -a){z-  p)  (z  -  y)"'(z  -  r)  =  0. 

But  z  is  different  from  a,  and  fi,  and  y,  etc.,  so  that 
none  of  the  binomial  factors  are  zero. 

.-.  a  =  0.  And  rejecting  aa;",  we  can  show  in  like 
manner  that  6  =  0;  thence  that  c  =  0,  etc.  And  the 
coefficients  being  severally  zero,  the  function  is  satisfied 
by  all  values  for  x,  since  it  is  zero  identically. 

196.   Let 

Ax'^+Bx'^-^+Cixf-^-] j-^=aa;"+6a;»-^+caj«-2H \-t. 

Then 

{A  -  a)x-  +  {B-  6)a;~-i  +  {C-  c)x--'-{- ...  +  ( 2"-  0  =  ^• 

231 


232  UKDETEKMINED    COEFFICIENTS. 

And  if  this  equation  is  to  be  true  independently  of 
the  value  of  x,  that  is,  for  all  values  of  x,  we  must  have 

A  =  a,  B=b,  C=C"'T=L 

And  this  establishes  the  principle  of  undetermined 
coefficients  for  functions  of  finite  dimensions. 

The  statement  of  the  principle  is,  that  if  a  positive 
integral  function  of  x  of  finite  dimensions  be  zero  for  all 
values  of  x,  the  coefficients  of  the  several  powers  of  x 
are  each  equal  to  zero. 

The  extension  to  functions  of  infinite  dimensions  will 
be  established  hereafter. 

We  shall  now  consider  applications  of  this  prolific 
principle. 

I.     Partial  Fractions. 


196.    The   sum  of  the  fractions    and  is 

■  . :  and  with  reference  to  this  latter  fraction,  the 
1  —  ar 

parts  which  make  it  up  by  addition  are  called  its  partial 
fractions.  It  is  often  necessary  to  separate  a  fraction 
into  its  partials,  it  being  understood  that  the  denomina- 
tors of  the  partials  shall  be  linear  whenever  practicable, 
but  at  any  rate  be  less  complex  than  that  of  the  original. 


Ex.  1.  To  separate      "*"  ^  into  its  partials. 
1  —  x'^ 

Since  the  denominator  is  (1  —  x)(l  -\-  x), 

.«i.Tr.o  S±x  _      A    ^      B     . 


l-x2      \-x     1  +  x' 
where  A  and  B  are  coefficients  to  be  determined. 


PARTIAL  FRACTIONS.  233 

Then,  S-\-x  =  A(l  -i-x)+  B{1  -  x). 

And  as  this  is  to  be  true  for  all  values  of  x,  we  apply  the  prin- 
ciple of  undetermined  coefficients,  which  gives 

S  =  A  +  B,  and  1  =  A  -  B, 

Whence  A  =  2,  and  B=l; 

and  3  +  x^_J_^.   1 


1—  X'^        1  —  X       1  -\-  X 

Ex.  2.   To  separate  into  its  partials. 

^  (X- l)(x-2)(x-3) 

Assume      ^ _=^_  +  ^_+_^. 

{x-l)(x-2)(x-S)      x-1     x-2     x-S 

Then 

x^  =  A(x  —  2)(x-S)-hB(x-l)(x-S)  +  C(x-l)(x-2). 

We  might  now  equate  coefficients ;  but  the  following  method  is 
simpler. 

Since  this  equation  is  to  hold  for  all  values  of  a;, 

Make  x  =  1 ;  then  1  =2A^    and  ^  =  J. 
Make  x  =  2;  then  4  =  —  5,  and  5  =  —  4. 
Make  x  =  S;  then  9  =  2  (7,    and  (7  =  f . 
/  x2  1  4.9 


(x  -  l)(aj  -  2)(x  -  3)      2  (X  -  1)      X  -  2      2  («  -  3) 

/v2  x  4-  1 

Ex.  3.  To  separate  — into  its  partials. 

(X-  l)2(x  +  2) 

In  forming  this  fraction  by  addition  there  may  have  been  a  frac- 
tion of  the  form  — - —  and  another  of  the  form  ,  and  in 

x-l  (x-  ly 

our  assumption  we  make  provision  for  these.    Therefore  we  assume 

x^-x-\-l      ^       A  B  C 

(x-l)2(xH-2)      {x-iy     x-l     x-{-2 

Then    x'^  -  x  +  1=  A(x  +  2)+  B (x  -  l)(x  +  2)  +  C (x  -  ly. 

Let  x  =  l ;  then  1  =  3^,  and  A  =  ^, 


234  PARTIAL   FRACTIONS. 

Substitute  J  for  A,  and 

x'i-ix  +  i  =  B(ix-  l)(x  +  2)+  C(x  -  1)2. 
Let  x  =  -2',  then  7  =  9  (7,  and  C  =  J. 

Substitute  |  for  C,  and  equate  the  coefficients  of  x^j  which  gives 

1  =  ^  +  J,  or  ^  =  f . 
x^  -  X  +  1       _         1  ,         2  7 


ix-\y\x  +  '2)      3(ic-l)2     9(ic-l)      9(a:4-2) 

3  'v^  I  3.  _  1 

I'x.  4.   To  separate  — i— H- ^ into  partials. 

X'^  —  1 

The  denominator  is  (x  —  l)(x^  -\-  x  +  1),  and  the  quadratic  fac- 
tor is  not  separable  into  real  factors. 

But  a  proper  fraction  with  a  quadratic  factor  in  its  denominator 
may  have  a  linear  factor  in  its  numerator.  We  make  provision 
for  this  by  assuming 

Sx'^  +  x-1  _  ^     ^    Bx  +  J7 


x^-l  x-l     x^  +  x  +  l 

Then,  3x2  +  X  -  1  =  A(x^  +  ^  +  1)  +  (^x  +  C)(a;  -  1), 

whence  we  readily  find      A  =  lj  B  =  2,  (7=2. 
.    3x2  +  x-l_      1    __^_  2x  +  2 


X^  —  1  X  —  1        X2  +  X  +  1 

For  a  fuller  discussion  of  this  subject  the  student  is 
referred  to  works  on  Higher  Algebra,  and  to  the  Calculus. 

EXERCISE  XV.  a. 

1.   Separate  into  partial  fractions  the  following  — 
ic  4-  2  .  ax  +  b 


(x  -  l)(x  -  2)  (a  -  x)(&  -  x)b 

oc+l  „  7x 


x2  _  5x  +  6  (2x  +  3)(x  +  2)2 

3x  —  2 .        ax 

(x-l)(x-2)(x-3)*  '  a2-x2' 


PARTIAL   FRACTIONS.  235 

197.  We  shall  now  extend  the  principle  of  undeter- 
mined coefficients  to  the  case  of  an  integral  function  of 
X  of  infinite  dimensions. 

Theorem.  In  a  positive  integral  function  of  x  of 
infinite  dimensions,  and  arranged  in  ascending  powers, 
any  term  may  be  made  greater  than  the  sum  of  all  that 
follow  by  making  x  sufficiently  small. 

Let  a -{- bx -{- coc^  -{- daP -i- ex^ -\- fx^  -\ be  the  function, 

and  let  ca^  be  the  term  chosen. 

Also  let  k  be  greater  than  any  coefficient  following  c. 

Then     kx^{l  -{-  x  -\-  x^  -\-  '")>dx^  +  ex^  -\-fx^  -f  ••• 

Or  kx^ '  -^—  >dx^-{-  ex^  -^fx^  +  ... 

1  —  x 

But        cx'>kj^'-^—  if  c>:^^ 


1—x  1—x 

And  since  — —  =  0  when  a;  =  0,  and  c  and  k  are  con- 
1  —  x 

stants,  — —  can  be  made  less  than  c  by  taking  x  suffi- 

1  —X 

ciently  small. 

.*.  cx^  can  be  made  >  dx^  +  ex'^  +fx^  H 

Now  let  A  +  Bx-\-Cx^-\ =  a  +  bx  +  cxr-\ be  true 

for  all  values  of  x.     Then 

ji^a  +  {B-b)x  +  (C-c)x^-\--*  =  0 

is  true  for  all  values  of  x. 

But  when  x  is  sufficiently  sm^ll,  ^  —  a  is  greater  than 
all  that  follows,  and  its  sign  controls  that  of  the  series ; 
but  the  whole  series  is  zero ;  therefore  A  —  a  =  0, 

,',  A  =  a. 


236  EXPA]srsTo:tT  of  functions. 

And  by  striking  out  A  and  a  as  being  equal,  we  prove 
in  like  manner  that  B=b'j  thence  C  =c,  etc. 

II.  .Expansion  of  Functions. 

198.  If  a  function  of  x  which  has  but  one  value  for 
each  value  of  x  be  expanded  in  powers  of  x,  it  must  take 
the  form 

a -{- bx -\- cx^ -}-  dx^  -\ 

where  every  exponent  is  a  positive  integer. 

For  if  there  be  a  term  of  the  form  gx"\  this  term  will 
become  infinite  when  x  =  0,  and  therefore  the  inde- 
pendent term  a  must  be  infinite,  and  the  expansion  is 
impossible.  ^ 

Again,  if  there  be  a  term  of  the  form  hx"",  this  term 
has  n  values,  and  therefore  the  expansion  has  at  least  n 
values  for  each  value  of  a;,  which  is  contrary  to  the 
hypothesis. 


Ex.  1.  To  expand 


I  —  X  —  x:^  -^  x^ 


Assume        ^  "*"  ^ — p^  =  I  +  ax -^  bx^ -]-  cx^  +  dx*  +  ... 
1  —  X  —  x^  -{-  x^ 


Then  I  +  x- Sx^  =  1  +  a\x  +  b 

-1       -a 


x'^  +  c 

~b 

1        -a 

+  1 
And  equating  coefficients, 


x^^d 
—  c 
-b 
+  a 


x^  +  . 


l  =  a  — 1,  —3  =  5  — a  —  1,  0  =  c— 6  —  a  +  1,  0  =  d— c  — &  +  a,  etc., 
whence  a  =  2,  6  =  0,  c  =  1,  c?  =  —  1,  e  =  0,  etc. 

And  the  expansion  is 

14-2x4-  0x2  4-x3-x*  +  0x5... 


EXPANSIOlSr   OF   FUNCTIONS.  237 

Ex.  2.  To  expand  Vl  +  x. 

Assume  y/\  -\-  x  =  1  +  «x  +  hx'^  +  cx^  +  c?x*  +  ••• 

Then         l+x  =  l  +  2aa:  +  25|aj2  +  2c    \x^-\-2d    »*  +  ■ 

aH  2ah\         2ac 

whence  a  =  },  6  =  —  J^,  c  =  jV,  <?  =  —  yf  j,  etc. 

.-.  vrTx  =  i  +  ix-ix^  +  j\x^-jijx^''' 

^         2-4  2.4.6  2.4.6-8 


EXERCISE  XV.  b. 

1.  Expand  ^(1  +2x  +  Sx'^  +  ix^  +  "-  wx«-i  +  •.-). 

2.  Expand  y/(l  +  x -{- x'^)  to  x*. 

3.  Expand  -jf^— ^)  to  x*. 

4.  Expand  ^(1  +  x)  to  x*. 

5.  Given    (1  +  a*)"  = ^—  =  1  +  CjX  +  CjX^  +  CgX^  +  ...,  to 

(1  +  x)   "^ 
find  the  coefficients  of  the  expansion  of  (1  +  x)-"  in  terms  of 
Ci»  C2,  c.^,  etc.,  up  to  the  fourth. 

6.  It  y  =  a^x  +  a.^x^  +  a^x^  +  •••,  find  x  in  terms  of  y. 
Assume  x  =  Ay  +  Bif  +  Cy^  +  •"  ;  write  for  y,  in  this  assump- 
tion, the  value  given,  and  equate  coefficients. 

7.  If  v  =  l4--  +  --  +  —  +  •••!    find  X  in  terms  of    z,  where 

/  1      2!      3! 

z  =  y  -  1. 

8.  If  X  =  ?/  -  2  ?/2  +  ?/3,  develop  ?/  as  a  function  of  x. 

9.  If    (a  +  7)x  +  rx2  +  ...)2  =  (a  +  2  6x  +  22cx2  +..-),   find    the 
values  of  a,  &,  c,  etc. 


238  SUMMATION   OF   SERIES. 

III.     Summation   of   Series. 

199.  It  will  be  noticed  in  Article  163  that  the  expres- 
sion for  the  nth  term,  as  a  function  of  n,  is  one  dimen- 
sion lower  than  the  expression  for  the  corresponding 
sum,  and  this  can  be  shown  to  be  true  for  all  series  of 
that  species. 

For  suppose     aS'„  =  a'nP  +  hn^~^  +  cn^~^  -\ 

Then    S,_^=  a{n  -  iy-\-  b(n  -  iy-^+  c{n  -  iy-'+  ... 

And  the  difference,  >S"  —  /S"~^,  is  the  nth  term ;  and  on 
expanding  n^  disappears. 

The  coefficient  of  n^"*  is  b  —  ap  —  b,  or  ap,  which  can- 
not be  zero  unless  a  or  p  is  zero,  both  of  which  suppo- 
sitions are  contrary  to  the  assumption. 

.-.  S''  —  aS"-^  is  of  the  (p  —  l)th  dimension. 

Ex.  1.  To  find  the  sum  of  the  series  of  squares  of  the  natural 
numbers,  viz.,  12  4-  22  +  3*^  +  •••  w^. 

Smce  the  nth  term  is  of  two  dimensions,  assume 

Sn  =  an^  +  hn^  +  en. 
Then         Sn-\  =  a(n  -  \y  +  h{n  -  1)2  +  c(n  -  1). 

.'.  8n  —  Sn-^1  =  3an2—  (3  a  —  2  5)w+a  —  &  +  c  =  ?ith  term 
=  n2. 

And  equating  coefficients, 

a  =  J,  5  =  J,  c  =  i. 

/.  Sn  =  in^  +  in'^  -^  in  =  "^(71  +  l)(2n  +  1). 
o 

Ex.  2.  To  find  the  nth  term,  and  the  sum  of  n  terms,  of  the  series 

1  +  4  +  8  +  14  +  23  +  30  +  .-. 


SUMMATION   OF   SERIES.  239 

Taking  first  differences,  we  have 

3  +  4  +  6  +  9  +  13  +  ... 

and  for  second  differences, 

1  +  2  +  3  +  4  +  ...,  an  A.  P. 

Now  as  the  nth  term  of  an  A.  P.  is  linear,  the  nth  term  of  the 
first  difference  is  quadratic,  and  of  the  original  series  is  cubic. 
Therefore,  assume  the  nth  term  =  a  +  6w  +  c/i^  -}-  dn^. 


When  n  =  l,  l  =  a+  6+  c+  d, 
n  =  2,  4  =  a  +  2&+  4c+  8d 
n  =  3,  8  =  a  +  36+  9c  +  27d. 
n  =  4,  14  =  a  +  46  +  16c  +  64d. 


3  =  6  +  3c+    Id. 

4  =  6  +  5c  +  19d 
6  =  6+7c  +  37df. 


l  =  6d     /.  (?  =  J. 


Thence  1  =  2  c  +  12  (^, 

2  =  2  c  +  18  (?. 

Thence  c  =  -  J,  6  =  %^,  a  =  -  2. 

And  the  nth  term  =  \{n^  -  3  n^  +  20  n  -  12). 

Next,  for  the  sum,  assume 

^„  =  an  +  hn^  +  cn^  +  (^n*. 

Sn-i  =  a(n  -  1)  +  6(n  -  1)2  +  cCn  -  1)^  +  d(n  -  1)*. 

Then    J (n^  -  3  n^  +  20  n  -  12)  =  i^n  -  Sn-\  =  the  nth  term 

=  a-6  +  c-(2+(2  6-3c  +  4d)n  +  (3c-6(2)n2  +  4  dn^. 

And  equating  coefficients,  we  get 

d  =  Jj,  c  =  -  ^\,  6  =  If,  and  a  =  —\%, 

.    Q  _  n{n3  -  2  n2  +  35  n- 10} 
•       "  24 


EXERCISE  XV.  C. 

1.    If  the  nth  term  of  the  series  l  +  8x+4x2+5x3+6a^+7x5+. 
is  of  tbe  form  1  +  ax  +  &x"^,  find  the  nth  term. 


240  MISCELLANEOUS. 

1  j_  2  x x'^ 

2.  Develop  — ,  and  show  what  two  series  it  is  the 

sum  of.  (l-x)(l  +  x)-^ 

(Separate  into  partial  fractions  and  develop  each.) 

3.  Sum  to  n  terms  the  series  whose  nth  term  is  1  —  n  +  n^. 

4.  Sum  the  series  whose  nth  term  is  |  (n^  +  n) . 

5.  Sum  to  ri  terms,  1 .  2  +  3 .  4  +  5 .  6  +  ••» 

6.  Sum  to  n  terms,  IJ  +  2  +  IJ  +  0  -  2J  ... 

7.  Sum  ton  terms,  1.2.3  +  2.3.4  +  3.4.5+ ... 

8.  Find  the  series  whose  nth  term  is  the  sum  of  the  natural 
numbers  from  1  to  n. 

IV.     Miscellaneous. 

200.  The  following  are  miscellaneous  applications  of 
the  principle  of  undetermined  coefficients  to  problems 
which  fall  under  none  of  the  previous  heads. 

Ex.  1.  To  find  the  condition  under  which  ax"-*  +  6x  +  c  shall  be 
a  complete  square. 

Assume  ax^  +  6x  +  c  =  (px  +  g)^. 

Then  expanding  (px  +  5)2,  and  equating  coefficients, 

a  =  p'^,    b  =  2pq^   c  =  q'^. 

But  (2i)g)2  =  4j)2g2. 

.-.  52  —  4:ac 
is  the  required  condition. 

Ex.  2.  To  find  the  condition  that  the  equation  x^-\-ax'^-{-bx-\-c=0 
may  have  two  equal  roots. 

This  is  the  condition  that  x^  +  ax^  +  6x  +  c  may  have  a  square 
factor. 

Assume    x^  +  ax^  +  6x  +  c  =  f  x  +  —  ]  (x  +  p)^. 

V     py 


MISCELLANEOUS.  241 


Expanding,  and  equating  coefficients, 
from  which  we  must  determine  p. 


1st 


9*i2  j_  C 


2  r)2    I    i^  —  ^29 

p  r«p2_26p4-3c  =  0, 

whence     {       „  ,      ^ 

^24.2c^^.  I  3p2_2ap+6  =  0. 

p 
Eliminating  linear  p,  p^  — 


3ac 

-62 

a^- 

35 

9c 

—  ah 

Eliminating  square  p,  -^  '    2ra2  -  3  6) 

...  4  (3  ac  -  62)  ((^2  _  3  6)  =  (9  c  -  a6)2 
IS  the  required  relation. 

Cor.  For  the    equation    x^  -\-  hx  +  c  =^  0^    we  get  by  making 
a  =  0,   46"^  —  27  c2,    as  the  condition. 

Ex.  3.   To  find  the  condition  that 

ax^  4-62/2  +  2  hxy  +  2  (/x  +  2./>/  +  c (A) 

may  be  the  product  of  two  factors,  rational  in  x  and  y  ;  and  to  find 
the  factors. 
Assume 
ax""-  +  by'^  +  2hxy  +  2 gx  +  2fy  +  c  =  (ax  +  ^  +  s\  (x  +  py  +  ■ 

and  equate  coefficients  of  xy  and  x,  and  we  obtain 

ap2  _  2  /ip  +  6  =  0,  and  s'^  —  2  gs  +  ac  =  0. 

Whence        p  =  -(h  -{■  V/i^  —  ab),  s  =  g  -^  Vg'^  —  ac. 
a 

Denote  Vh^  —  ab  by  H,  and  ^/g'^  —  ac  by  G,  and  the  factors 
become 

{ «^  +  rz^+ ^  + '^  I  ^  +  ^^^^"^^ + ^1' 

I  h  +  H  Jl  a  g  -\-GfJ 

or  i{ax  +  (;i  -  JET)?/  +  ^  +  G'}{ax  +  (/i  +  ^)?/  +  ^  -  G^}, 

a 


242  MISCELLANEOUS. 

As  these  factors  do  not  contain  /,  we  equate  the  coefficients  of 
linear  y  from  the  function  and  from  its  factored  equivalent,  and 
obtain 

2/=  1{(^  ^G){h-^H)  +  {g-  G)(h  -  IT)}  =  ?  (gh  +  GH). 
a  a 

And  putting  for  G  and  H  their  values,  and  reducing,  we  obtain 

ahc-\-2fgh-aP-hg'^-ch'^  =  () {B) 

which  expresses  the  necessary  condition. 

This  very  important  function,  B,  is  called  the  dis- 
criminant of  the  function  A, 

Ex.  4.  To  find  a  number  such  that  if  1  be  added  to  it  the  sum 
shall  be  a  square,  and  if  1  be  subtracted  from  it  the  difference  shall 
be  a  square. 

Let  X  denote  the  number. 

Then  ic  +  1,  and  x  —  1,  and  consequently  x^  _  |^  are  all  to  be 
squares. 

Assume  x'^  —  \  ={x  —  pY  =  x"^  —  2px  ■\- p\ 

Then  x  =  l-±^, 

2p 

and  ^+1^(1 +P)^, 

2p 

which  will  be  a  square  if  2  p  is  a  square. 

Let  2p  =  s2  ;  then  p  =  -,  and  x  =  i±-^; 

where  s  may  be  any  quantity  whatever. 
When  s  =  \,\,  1,  2,  3,  4... 

and  the  problem  is  thus  indeterminate. 


EXERCISE  XV.  d. 

1.  Find  the  condition  that  x'^  —  ahx-\-  he  may  be  a  complete 
square. 


MISCELLAKEOUS.  243 

2.  If  x^  +  ax2  +  6a;  +  c  is  a  complete  cube,  show  that  27  c  =  a^, 
and  3  &  =  a^. 

3.  Find  the  condition  that  ax^  -\-  bx  +  c  and  a^x^  +  b^x  +  Cj 
may  have  a  common  linear  factor. 

2  A        X        2  B 

4.  Determine  X  so  that  the  equation 1 1 =  0  may- 
have  equal  roots  in  a;.                               x-^a     x     x-a 

6.  Show  that  1 +^  +  i?^  +  i^^+ ...   is  the  square  of 
12!  3!  ^ 

6.  Find  the  value  of  m  that  y  —  mx  —  S  =  0  may  be  compati- 
ble with  y  —  X  —  1  =0  and  y  —  2x  —  2  =  0. 

7.  Find  the  value  of  c  in  order  that  2x^-}-y'^  —  4:xy  +  6y  +  c 
may  be  put  into  rational  factors  in  x  and  y,  and  find  the  factors. 

8.  Show  that  ax^  +  2  hxy  +  ftz/^  can  always  be  rationally  fac- 
tored, whatever  be  the  values  of  a,  h,  and  b  ;  and  find  the  factors. 

9.  Find  a  formula  for  numbers  which  put  for  x  make  x"^  -\-  b  a> 
complete  square. 

10.  Determine  the  fraction  of  form  — ^  "^      — ,  which  expands 

1  +  ex  +  dx'^  ■ 

into  1  +3x  + 4x2+7^3  +  llx*+  18x^  +  29x6  +  ... 

11.  Find  the  relation  between  a  and  b  that  (x  —  a)2  +  (x  —  6) 
may  be  a  square. 

12.  Express  x*-4  x3H-x2+2  x  in  the  form  (x2+ax+  6)2-  C^+c)2 ; 
and  thence  show  how  the  corresponding  equation  can  be  solved. 

13.  Put  x2  +  x?/-2?/2  4-2x  +  72/-3  into  factors. 

14.  Find  the  value  of  m  that  2x2  —  3xy4-2x  —  y  +  m  may  be 
put  into  rational  factors  in  x  and  y. 

15.  If  ax2  +  6?/2  +  2  hxy  -\- 2  gx  +  2  fy  -\-  c  is  expressible  in  fac- 
tors rational  in  x  and  ?/,  show  that  h  has  two  values  for  given  val- 
ues of  a,  6,  /,  ^,  and  c,  unless  /-  =  6c,  or  g'^  =  ac. 


CHAPTER   XVI. 
Elementary  Continued  Fractions. 

201.    Take  any  proper  fraction,  preferably  in  its  lowest 
terms,  as  ^. 

Then, 
11^  l^l^l^l^  1 

25     25     2  + A     2  +  —     2H —     2+        ^ 


11         '11         'll  3^2  3^     1 


This  expanded  result  is  a  continued  fraction,  and  is 
often  written 

1 

2  +  ^-      1 

2 

Or,  for  the  purpose  of  saving  space,  it  is  more  generally 
written 

1111 

2  +  3  +  1+2* 

From  the  nature  of  the  expansion,  it  is  evident  that 
every  fraction  can  be  expressed  as  a  finite  continued 
fraction. 

Ex  ^  =  '^     1     1     1     1. 

Ill      2  +  1+3  +  2+4 

244 


CONTINUED   FRACTIONS. 


245 


202.  Problem.  To  express  any  vulgar  fraction  as  a 
continued  fraction. 

Proceed  as  in  finding  the  G.  C.  M.  of  the  numerator 
and  denominator  of  the  given  fraction,  and  write  the 
quotients  in  order  as  denominators  of  the  continued 
fraction,  the  numerators  being  1. 

Ex.  1.   To  change  j\%  to  a  continued  fraction. 


56 

103 

11 

9 

47 

1 

1 

2 

5 

4 

2J 

-  quotients.     .-.  — - 


56_. 
103" 


11111 

1  +  1  +  5  +  4  +  2* 


Ex.  2.    To  change  3.1416  to  a  continued  fraction. 

This  is  3  +  ^^^1  and  we  change  xVoVo  ^^  ^  continued  fraction. 


1416  I  10000 


7 
16 
11 


1416  . 

10000  ■ 


1      Jl      i_ 

7  +  16  +  11' 


and  3.1416  =  3  +  - 


1       1 


Ex.  3.   To  express 


X2  +  X  +  1 


a;2  +  X  +  1  _  1      a;+  1 

X2  11 


7  +  16  +  11 
as  a  continued  fraction, 

—  X  .   X  +  1  _ 


-^ —  =  x  + , 

5C+  1  X+  1 


X  X 


X'^  +  X+l        1  +  X+  (-1)+  (-X) 

203.    The  quotients  obtained  by  the  process  of  finding 
the  G.  O.  M.  are  called  partial  quotients. 

Let  the  partial  quotients  be  denoted  by  aj,  ag,  a^,  etc., 
all  the  quantities  being  affected  with  the  sign  +. 
Then  the  continued  fraction  is 

ill  1 

ai  +  ag  +  agH ha„' 


246  CONTINUED  FRACTIONS. 

which  in  its  totality  is  equal  to  the  fraction  from  which 
it  is  derived,  and  which  we  shall  denote  by  x. 

—  is  the  1st  convergent,  which  denote  by  Vi ; 

—  —  is  the  2d  convergent,  which  denote  by  Vj  5 
tti  -f  ttg 

—  —     —  is  the  3d  convergent,  v^ ; 
ai  +  ag  +  «3 

etc.  etc.  etc. 

Then 

—  >x,  since  its  denominator  a^  is  too  small. 

Vg?  or  —     —  <x,  since  its  denominator  is  too  great. 

«!  +  ^2 

Vg,  or  —      —     —  > cc ;  ^4  or  —      —      —      —<x) 

«!  +  «2  +  «3  «!  +  ^2  +  tta  +  ^4 

etc.  ••• 

Thus,  all  the  odd  convergents  are  greater  than  x,  and 
all  the  even  convergents  are  less  than  x\  so  that  the 
value  of  X  lies  between  that  of  any  two  consecutive  con- 
vergents, until  we  reach  the  last  convergent,  which  is  the 
value  of  X  itself. 

Illustration.  — The  convergents  to  \\  (Art.  201)  are 

Vl  =  J,    ^2  =  f ,    V3  =  1,    n=l\  =  «. 

Now         \\  =  x  =  0.44. 

Vj  =  0.50,  and  is  too  great  by  0.06. 

^2  =  0.42857  .-.  and  is  too  small  by  0.02142  ... 

Vj  =  0.4444  •••  and  is  too  great  by  0.0044 ••• 


CONTINUED   FllACTIONS.  247 

Thus  the  consecutive  convergents,  while  being  alter- 
nately too  great  and  too  small,  approximate  more  and 
more  nearly  to  the  value  of  a?;  and  hence  the  names 
convergent  and  converging  fractions. 

We  thus  see  that  one  obvious  application  of  continued 
fractions  is  to  find  a  fraction,  with  few  figures,  which 
shall  be  a  close  approximation  in  value,  to  a  given 
fraction  with  so  many  figures  as  to  be  unwieldy. 

Thus  f  and  Jf  are  close  approximations  to  f fif. 

204.  The  first  convergent  Vi  =  —,  and  the  second, 
V2  = ;  so  that  to  get  the  second  convergent  from 

the  first  we  write  ai  H — ■  for  ai  in  the  first. 

Similarly,  to  get  v^  from  V2  we  write  a2-\ —  for  ag  in 


a 


the  value  of  V2;  and  generally,  to  get  v„^i  from  v„,  we 
write  a„  H for  a^  in  the  value  of  i;„. 

P  P  P 

Now  let  Vi  =  — ^  V2  =  -?,  ^3  =  -?,  etc. 
Vi  ^2  Vs 

Then       Vi  =  -^  =  —',  V2  =  -^  = 


^      Q3     a3(a2ai  +  l)  +  «i      a3Q2+Qi' 


-^4  = 


(^a3  +  ^y2  +  P, 
P4  _  V  CI  J 


'         ^^         (-3  +  ^)&+Q.  ^^^^+«^ 


etc. 


248  CONTINUED   FRACTIONS. 

We  here  see  that  the  forms  for  %  and  v^  are  exactly 
alike,  the  only  difference  being  that  the  subscripts  are 
each  increased  by  unity. 

To  prove  that  this  is  always  the  case. 

Assume  v,,  =  ^  =  ^^^^-'  +^-2. 


Qn         CtnQn-1  +  Qn-2 


an  +  —  ]Pn-l+Pn- 


Then    v      .  _  ^n+i  _  V  «n+i. 

_  an+l(<^nPn-l  +  Pn-2)  +  Pn-l  _,  ^n+l^n  + -Pn--1. 

Which  shows  that  the  form  is  true  for  v^+i  if  it  is  true 
for  Vn'  But  it  is  true  for  v^  and  V4,  and  therefore  for 
Vsj  Vqj  etc. ;  i.e.  it  is  generally  true. 

205.  The  result  of  the  preceding  article  furnishes  a 
convenient  means  of  finding  all  the  consecutive  con- 
vergents,  when  we  have  any  two  consecutive  ones,  and 
the  partial  quotients. 

P       P 

Taking  _?i=l^  —^^  a„+i,  we  get  the  (n  +  l)th  convergent 

Qn-l     Qn 

as  follows : 

Multiply  P„  by  a„^i  and  add  P„_i,  for  P„^i ;  and  mul- 
tiply Q„  by  a,+i  and  add  Q„_i,  for  Q„+i. 

This   operation   gives  ^^  =  ^^+^^^  +  Pn-i.    ^hich   is 

correct.  ^"+1         <^n+lQn+6n-l 

Eor  convenience  we  assume  a  fictitious  convergent, 
Vq  =  ^,  and  carry  out  the  operation  as  in  the  following 
example : 


CONTINUED   FRACTIONS.  249 


Ex.   Let  2,  1,  3,  1,  2  be  partial  quotients. 

0  1         i ^ 6_U 

1  2         1         '^         1        2 


11       14      39 


The  partial  quotients  after  the  first  are  written  between 
the  lines,  and  the  parts  of  the  corresponding  convergents 
are  written  above  and  below  the  lines. 

Thus  the  convergents  are 

Vi  =  h  '^2  =  h  ^3  =  IT*  n  =  T4>  and  ^5  =  if. 
206.    Taking  the  convergents  of  the  preceding  example, 
111  14-1 

v,-v,=---= ]    V,,-Vo  =  --  —  ^ 


2      3      2x3'        ^      3      11      3  X  11 

4        5            1  ^.^ 

v«  —  v.= = :  etc. 

^        *      11      14      11  X  14  ' 

Thus  the  difference  between  two  consecutive  conver- 
gents is  the  fraction  whose  numerator  is  1,  and  whose 
denominator  is  the  product  of  the  denominators  of  the 
two  convergents.  And  this  difference,  taken  in  regular 
order,  is  alternately  positive  and  negative. 

To  prove  that  this  is  always  the  case. 

But,  Art.  204, 

=  —  {■Pn-lQn-2  —  Qn-l^n-2)' 

Similarly, 

-  {Pn-lQn-2  -  Qn-lPn-2)=Pn-2Qn-S  "  Q«-2^n-3  =  etC, 

SO  that        PnQn-i  —  QnPn-i        cvidcutly   has,   with   the 
exception  of  sign,  the  same  value  for  all  values  of  n. 


250  CONTINUED   FRACTIONS. 

But  when  n  =  2,  P2Q1  —  Q2P1  =  ^2%  —  (^A  +  1)  =  —  1 5 
and  for  71  =  3,  P3Q2  -  QsA  =  +  1- 


Hence  ?;„  —  v^-i  = 


(-1)"-', 

Q„Qn-l 


Which  proves  the  theorem.  And  as  the  difference 
between  any  two  consecutive  convergents  is  a  fraction 
with  unity  as  numerator,  every  convergent  is  in  its  low- 
est term,  i.e.  its  parts  are  prime  to  one  another. 

207.   As  the  value  of  x  lies  between  those  of  two  con- 

P 

secutive  convergents,  —~  differs  from  x  in  value  by  less 

1  Qn  1 

than  — -— — ,  i.e.  by  less  than  -— . 

QnQn+l  Qn 

Thus  y*i,  a  convergent  to  |f ,  differs  from  the  latter  fraction  by 
less  than  y^y  ;  and  ^s^,  another  convergent  to  the  same,  differs  from 
it  by  less  than  jj^. 

Kow  if  a„  is  relatively  large,  the  quantities  —  Pn-29 
and  — .  Q„_2  are  relatively  small,  and  the  whole  fraction 
differs*  but  little  from  ^^. 

Qn-l 

That  is,  if  a„  is  relatively  large,  the  difference  between 
v^_i  and  Vn  is  relatively  small,  and  hence  v^-i  is  a  close 
approximation  to  x. 

Hence,  the  last  convergent  preceding  a  large  partial 
quotient  is  a  close  approximation  to  the  value  of  the 
fraction. 


CONTINUED   FRACTIONS.  251 

Thus,  if  the  partial  quotients  be 

1,  2,  1,  3,  15,  2, 
the  convergents  are 

Vj  =  1,  1^2  =  2,  v^  =  1,  V,  =  \\,  Vg  =  Iff,  vg  =  a;  =  f f  J  ; 

and  v^,  preceding  the  quotient  15,  differs  from  x  by  less  than 

^         or-^. 
15  X  229        3435 

EXERCISE  XVI.  a. 

1 .  Find  all  the  convergents  to  Jf  f . 

P  a 

2.  If  -  be  the  convergent  preceding  -,  show  that  Ph  differs 

q  b 

from  Qa  by  unity. 

Thence  form  a  rule  for  finding  multiples  of  two  numbers  prime 
to  one  another,  so  that  such  multiples  shall  differ  by  a  given  integer. 

3.  Find  multiples  of  23  and  31  that  shall  differ  by  6. 

4.  Find  an  improper  fraction  to  express  3.1416  to  a  near 
approximation. 

6.  Express  1.4142  by  a  vulgar  fraction,  each  of  whose  parts  are 
less  than  100. 

209.  A  continued  fraction  may  be  non-terminating; 
i.e,  its  partial  quotients  may  be  an  endless  series  of 
numbers. 

The  convergents  approximate  in  the  same  way  whether 
the  fraction  is  terminating  or  not,  but  no  convergent, 
however  high  its  order,  can  express  exactly  the  quantity 
denoted  by  the  continued  fraction.  Such  a  fraction  has 
in  general  an  incommensurable  for  its  value. 

If  the  partial  quotients  exist  in  recurring  periods,  like 
the  figures  in  a  circulating  decimal,  the  fraction  is  a 


252  CONTINUED   FKACTIONS. 

periodic  continued  fraction,  and  every  such  fraction  is  the 
development  of  a  square  root  or  quadratic  surd. 

An  infinite  continued  fraction,  in  which  the  partial 
quotients  are  not  periodic,  may  be  the  expansion  of  a 
cubic  or  higher  form  of  surd  expression,  but,  in  general, 
the  equivalent  surd  expression  cannot  be  found. 

Ex.  1.  Let  the  partial  quotients  be  2,  2,  2  •••,  and  let  x  be  the 
equivalent  fraction. 

Then  a.  =  l     1     1         ~     ' 


2  +  2  +  2+  2-\-x 

.'.  x2  +  2  X  =  1,  and  ic  =  V^  -  1- 

Ex.  2.   Let  the  partial  quotients  be  1,  2,  3,  1,  2,  3  ••• 

,     Then  x  = 1 .  =  1±^, 

l+_i_      10  +  3x 

2+-i- 

S  +  x 

.-.  3a:2  +  8x  =  7,  and  X  =  K">/37  -  4). 

And  the  method  applies  to  all,  however  great  the 
periodic  part  may  be. 

210.  To  expand  the  square  root  of  a  non-square  num- 
ber into  a  periodic  fraction. 

We  give  the  method  of  operation  by  means  of  examples. 

Ex.  1.    To  develop  ^1. 

Since  2  is  the  highest  integer  in  the  root  of  7,  we  subtract  2  from 
^7,  and  throughout  the  operation  no  number  greater  than  2  is  to 
be  thus  subtracted. 

3 


7  =  2+ V7 -2  =  2  + 
multiplying  ^/l^  by  ^^. 


V7  +  2' 


V7  +  ^^i  + 

3 


CONTINUED   FEACTIONS 

2 


253 


^^1^  =  1+- -, 

3  V7  +  1 


adding  1  to  2  to  get  an  integral  quotient,  1,  and  subtracting  1 
from  y/1  so  as  to  keep  the  whole  unchanged. 

2  2  V7  +  1' 

3  ^3  V7  +  2' 
V'^  +  ^  =  4  +  ^7  _  2,  etc. 

The  partial  quotients  obtained  are  2,  1,  1,  1,  4;  and 
as  we  have  now  to  begin  again  with  ^1  —  2,  the  same 
quotients  will  constantly  recur.  We  notice  then  that 
when  a  quotient  which  is  double  the  first  one  appears, 
the  period  is  complete. 

^  ^1+1+1+4+ 


Ex.  2. 


VIS  =  3  +  V13  -3  =  3  + 


V13  +  3 

a/13  4-3^.^       V13-l^.  3 

4  "^4  V13  +  1 


V13  +  1^1   1   V^-^- 


■  =  1  + 


V13  +  2 

Vl3  +  2^         V13-1^  4 

3  3  V13  +  1 

4  ^4  ^Vl3  +  3 
V13  +  3  =  6+  V13  -  3,  etc. 

.,^13  =  3  +  1     1      1     1     1      ... 


254  CONTINUED   FRACTIONS. 

The  convergents  to  the  fractional  part  are 

Vj  =  1,  ^2  =  h  ^3  =  h  v^  =  f ,  etc. 
/.  the  convergents  to  ^13  are 

4,  3},  3|,  3|,  3ff,  etc. 

EXERCISE  XVI.  b. 

1.  Find  the  value  of  1  +  -      -     -     i  — 

2+3+2+3 

2.  Find  the  value  of  1     -     -     -     L. 

1+2+3+4+1 

8.  Show  that  the  C.  F.  whose  partial  quotients  are  1,  —  2,  3, 
—  4,  forming  a  period,  has  3  for  its  total  value,  and  find  the  first 
10  convergents. 

4.  Find  the  value  of  2  +  -     -     -     -      ... 

2+4+2+4+ 

6.  Expand  ^2  and  y^6  into  periodic  C.  F.'s. 
6.  Expand  y'l?  and  y'19  into  periodic  C.  F.'s. 


CHAPTER  XVIL 
Logarithms  and  Exponentials. 

211.  In  the  expression  a^=h,  x  is  called  the  logarithm 
of  h  to  the  base  a;  and  this  relation  is  otherwise  indi- 
cated by  writing  x  =  log„&. 

The  base,  a,  being  some  fixed  positive  number,  to 
every  value  of  b  there  is  a  corresponding  value  of  x. 

If  these  corresponding  values  be  tabulated  in  opposing 
columns,  the  6-column  is  a  column  of  numbers,  and  the 
ic-column  is  a  column  of  logarithms,  and  the  whole  forms 
a  table  of  logarithms  to  the  base  a. 

As  will  be  shown  hereafter,  the  general  properties  of 
logarithms  are  the  same  for  all  bases,  and  any  positive 
number,  commensurable  or  incommensurable,  may  be 
taken  as  a  base;  but  certain  numbers  offer  special  ad- 
vantages as  bases  in  working  with  logarithms,  and  in 
calculating  them. 

As  a  consequence  logarithms  are,  in  practice,  taken 
to  one  of  two  bases ;  namely,  10,  as  being  the  base  of 
our  numerical  system ;  and  a  certain  incommensurable, 
usually  denoted  by  e,  and  called  the  Napierian  or  natural 
base. 

Logarithms  to  the  base  10  are  decimal  or  common 
logarithms,  and  those  to  the  base  e  are  Napierian  or 
natural  logarithms. 

255 


256        GENERAL   PROPERTIES    OF   LOGARITHMS. 

GENERAL    PROPERTIES    OF    LOGARITHMS. 

212.    Let  a"  =  h,  and  a^  =  c. 

Then  x  =  log  J),  and  y  =  log„c. 

(1)  bc  =  a''a^  =  a'+^, 
and               log^(bc)  =  X  +  y, 

or  log„  (&c)  =  log„6  +  log„c. 

That  is,  the  logarithm  of  the  product  of  two  numbers 
is  the  sum  of  the  logarithms  of  the  numbers. 

(2)  ^  =  ^  =  a^-y. 


log/-j=a;-2/, 


or  log„/^- j  =  log,6  -  log„c. 

That  is,  the  logarithm  of  the  quotient  of  two  numbers 
is  the  logarithm  of  the  dividend  diminished  by  the  loga- 
rithm of  the  divisor. 

(3)  (a-)"  =  6-=  a--. 

.*.  loga(&")  =  nx  =  nlogj). 

That  is,  the  logarithm  of  the  nth  power  of  a  number  is 
n  times  the  logarithm  of  the  number. 

(4)  Writing  -  for  n, 

n 

log.(&»)=-logA 

n 


EXPONENTIAL  EQUATIONS.  257 

or  the  logarithm  of  the  nth  root  of  a  number  is  one-nth 
of  the  logarithm  of  the  number. 

These  four  relations  form  the  working  rules  of  loga- 
rithms in  their  applications  to  quantity. 

213.  The  results  of  the  preceding  article  show  that 
multiplication  in  numbers  corresponds  to  addition  in 
logarithms  ;  division  in  numbers,  to  subtraction  in  loga- 
rithms ;  the  raising  of  a  number  to  a  power,  to  the  mul- 
tiplication of  a  logarithm  by  a  number ;  and  the  extracting 
of  the  root  of  a  number,  to  the  division  of  a  logarithm  by 
a  number. 

There  are  in  arithmetic,  as  confined  to  numbers,  no 
known  operations  which  correspond  to  the  multiplication 
or  division  of  one  logarithm  by  another,  and  hence  to 
the  raising  of  a  logarithm  to  a  power,  or  to  the  extrac- 
tion of  its  root. 

Such  operations  upon  logarithms  can  correspond  only 
to  some  hyper-arithmetical  processes.  Thus  logarithms 
not  only  facilitate  the  more  difficult  arithmetical  opera- 
tions ;  they  also,  by  an  extension  of  processes,  give  rise 
to  a  sort  of  transcendental  arithmetic. 

EXPONENTIAL    EQUATIONS. 

214.  An  exponential  equation  is  one  in  which  the 
variable  appears  as  an  exponent. 

Thus  a'  =  b,  with  x  variable,  is  an  exponential  equation. 
The  method  of   solution  is  obvious ;   for  taking  the 
logarithms  of  both  members, 

log(a'=)  =  a^loga  =  log  6. 

loga 


258  EXPONENTIAL   EQUATIONS. 

And  the  operation  which  gives  x  is  the  transcendental 
one  which  corresponds  to  the  division  of  one  logarithm 
by  another. 

Ex.  1.   Given  a''  +  a-'  =  26;  to  find  x. 

Multiply  by  a*, 

a^x  -2ba'  +  1=0. 

Whence  a'  =  b±  Vb^  - 1,  (Art.  126) 

and  ^^logjb+Vb^^l 

log  a 

Ex.  2.  To  express  the  logarithm  of  \'  ^  ' — ^,  in  terms  of  the 
logarithms  of  2,  3,  5,  and  7.  2^(216)* 

5.9 

The  given  expression  reduces  to  (3i ^  •  7^)  h-  (2^3 ,  56^ .  and  hence 
its  logarithm  is 

fflog3  +  61og7  -  131og2-61og5. 

EXERCISE  XVII.  a. 

1.  If  3  be  taken  as  a  base,  what  are  the  logarithms  of  9,  of  81, 
of  729,  of  2V,  of  ^^3  ? 

2.  If  6  be  the  base,  show  that  |  is  the  logarithm  of  14.697. 

3.  If  J  be  the  base,  of  what  numbers  are  1,  2,  0,   —  -, 

the  logarithms  ?  ^ 

4.  Prove  that,  with  any  positive  base,  1  is  the  logarithm  of  the 
base,  0  is  the  logarithm  of  1,  and  —  00  is  the  logarithm  of  0. 

5.  If  logfc  a  =  n  loga  6,  show  that  logfe  a  =  ^  '  ^^  ,  where  the 
logarithm  without  a  suffix  is  taken  to  any  base.     ^^  ^ 

6.  Solve  the  exponential  equations  — 

i.   20^  =  100.  iv.   oc^  =  y'^  and  x^  =  y\ 

ii.    (23)a:.(32)«  =  4.9.  V.    2a^+  a^^  =  a^, 

iii.   32*.  53*-^  =  7=^-1112-*.       vi.   ai.a2.a3...Qjx_yj, 


OF  THE   TABLE   OF   LOGARITHMS.  259 

7.  Express  the  logarithm  of  (8  V3  •  ^12)  ^  (  V^  •  v^l^)  in  terms 
of  the  logarithms  of  2,  3,  and  6. 

8.  Given  log  2  =  0.30103  ;  find  log  64,  log  256,  log  vl28,  log  J, 
log  25.     (Base  =  10.) 

9.  Express  in  terms  of  log 2  and  log  3  the  logarithms  of  6,  18, 
72,  ji^,  0.25,  0.0416. 

10.  How  many  terms  of  the  G.  P.   1  +  §  +  |  +  •••  will  make 

18915  9 

11.  How  long  will  it  take  a  sum  of  money  to  double  at  5% 
compound  interest  ? 

OP    THE    TABLE    OP    LOGARITHMS. 

216.  In  a'  =  b,  iib  is  greater  than  a  and  less  than  a% 
0?  is  >  1  and  <  2 ;  i.e.  oj  =  1  +  a  proper  fraction. 

If  6  is  >  a^  and  <a^,  a?  =  2  +  a  proper  fraction ;  etc. 

Thus  a  logarithm  consists  of  an  integral  part,  called 
the  characteristic,  and  a  fractional  part,  called  the  man- 
tissa.    Either  of  these  may,  however,  become  zero. 

Taking  10  as  a  base,  every  integral  power  of  10  con- 
sists of  1  followed  by  ciphers  only,  and  the  logarithm  of 
such  power  is  an  integer,  or  characteristic,  being  the 
index  of  the  power. 

Thus  log  100  =  2,  log  1000  =  3,  etc. 

For  any  number  between  100  and  1000  the  logarithm 
is  2  +  a  decimal ;  for  a  number  between  1000  and  10000, 
it  is  3  +  a  decimal ;  etc. 

Hence  one  convenience  of  decimal  logarithms  is  that 
we  know  the  characteristic  at  sight,  and  it  is  not  nec- 
essary to  tabulate  it. 

The  following  rule  gives  the  characteristic  for  decimal 
logarithms : 


260  OF   THE   TABLE   OF   LOGARITHMS. 

Call  the  units'  place  of  the  number  zero,  and  count  to 
the  significant  figure  farthest  upon  the  left ;  the  number 
of  this  figure  is  the  characteristic,  positive  if  counted 
leftward,  and  negative  if  counted  rightward. 

Thus  the  characteristic  of  the  logarithm  of  0.000074  is 

-  5,  of  386.50  it  is  2,  and  of  430070  it  is  5. 

216.  The  Mantissa.  Let  n  be  a  number,  and  let  c  and 
m  be  the  characteristic  and  mantissa  of  its  logarithm. 

Then  log  n=:zc-\-m. 

To  divide  n  by  10*  we  subtract  log  lO''  from  log  n.  But 
log  10"=  =  07;  and  dividing  a  number  by  an  integral  power 
of  10  has  no  effect  other  than  moving  the  decimal  point. 

Therefore  log(n-^  10'')  =  (c  —  a?) +  m, 

and  since  x  is  integral,  the  mantissa  is  unchanged. 

Hence  the  mantissa  of  a  logarithm  to  base  10  does  not 
depend  upon  the  position  of  the  decimal  point,  but  only 
upon  the  arrangement  of  figures  in  the  number ;  so  that 
the  same  arrangement  always  corresponds  to  the  same 
mantissa,  and  vice  versa. 

The  characteristic,  on  the  other  hand,  is  determined 
wholly  by  the  position  of  the  decimal  point. 

Thus  the  logarithms  of  0.0024,  0.24,  240,  24000,  etc., 
all  have  the  same  mantissa,  while  the  characteristics  are 

—  3,  —  1,  2,  and  4  respectively. 

217.  As  the  logarithms  of  integral  numbers  are  mostly 
incommensurable,  the  approximation  to  their  mantissse 
is  carried  to  4,  5,  6, 7,  etc.  decimal  places,  thus  giving  rise 
to  tables  of  4-place,  5-place,  6-place,  or  7-place  logarithms. 


OP   THE   TABLE   OF   LOGARITHMS. 


261 


Portions  of  a  table  of  5-place  logarithms.  A,  from  number  1780 
to  1889  ;  B,  from  number  5700  to  5769 ;  and  C,  from  number  7320 
to  7429. 


N. 
178 

0 

1 

2 

3 

4 

5 

6  1  7 

8 

9 

D. 

25042 

066 

091 

115 

139 

164 

188: 212 

237 

261 

9 

285 

310 

334 

358 

382 

406 

431 

455 

479 

503 

24 

180 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

I 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

2 

26007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

3 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 

4 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

5 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

6 

951 

975 

998 

021 

045 

068 

091 

114 

138 

161 

23 

7 

27184 

207 

231 

253 

277 

300 

323 

346 

370 

393 

8 

416 

439 

462 

485 

508 

531 

554 

577 

600 

'"'. 

570 

75587 

595 

603 

610 

B. 

618 

626 

633 

641 

648 

656 

I 

663 

671 

679 

686 

694 

702 

709 

717 

724 

732 

2 

740 

747 

755 

762 

770 

778 

785 

793 

800 

808 

3 

815 

823 

831 

838 

846 

853 

861 

868 

876 

884 

8 

4 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 

5 

967 

974 

982 

989 

997 

005 

012 

020 

027 

035 

6 

76042 

050 

057 

065 

072 

080 

087 

095 

103 

no 

732 

.86451 

457 

463 

469 

C. 

475 

481 

487 

493 

499 

504 

3 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 

4 

570 

576 

581 

587 

593 

599 

605 

611 

617 

623 

5 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 

6 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 

7 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

6 

8 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

9 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

I 

982 

988 

994 

999 

005 

on 

017 

023 

029 

035 

2 

87040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

r24 

2 

5 

7 

10 

12 

14 

17 

19 

22 

P.  \    g 

2 

5 

7 

9 

II 

14 

16 

18 

21 

I 

2 

2 

3 

4 

5 

6 

6 

7 

I  6 

I 

I 

2 

2 

3 

4 

4 

5 

5 

262  OF   THE   TABLE   OF   LOGARITHMS. 

The  larger  tables  are  mostly  7-place,  but  5-place  loga- 
rithms are  sufficiently  accurate  for  the  majority  of  arith- 
metical applications. 

We  give  on  page  261,  and  merely  for  purposes  of  illus- 
tration, portions  of  a  5-place  table,  in  which,  as  is  usual, 
only  mantissse  are  registered. 

218.  The  working  a  table  of  logarithms  consists  in  two 
operations  inverse  to  one  another ;  namely, 

(a)  to  find  the  mantissa  corresponding  to  a  given 
arrangement  of  figures  in  a  number,  and 

(h)  to  find  the  arrangement  corresponding  to  a  given 
mantissa. 

(a) 

A  complete  5-place  table  gives  the  mantissse  for  every 
arrangement  of  4  figures  from  1000  to  9999 ;  the  three 
right-hand  figures  being  taken  from  column  N,  and  the 
fourth  from  the  horizontal  line  at  the  top  of  the  table. 

Thus,  for  1854  the  mantissa  is  26811 ;  for  1864  it  is 
27045,  the  last  three  figures  045  being  in  distinctive  type 
to  show  that  the  first  two  figures  of  the  mantissa  are  to 
be  taken  from  the  first  column  and  the  line  below,  being 
27  instead  of  26. 

Ciphers  occurring  before  or  after  an  arrangement  do 
not  affect  the  mantissa. 

Thus,  23,  2300,  23000,  .023,  etc.,  have  the  same  man- 
tissa. 

Ex.  1.  To  find  the  mantissa  of  18347. 

Mantissa  for  18340  is  26340. 
Mantissa  for  18350  is  26364. 
Difference  10  24. 


OF   THE   TABLE   OF   LOGARITHMS.  263 

Thus  each  unit  between  18340  and  18350  adds  f  ^  to  the  man- 
tissa, and  hence  7  adds  7  x  f  J,  or  17  nearly. 

.-.  26340  +  17  =  26357  is  the  mantissa  required. 

The  column  marked  D  (differences)  and  the  row  at  the  bottom 
marked  P  (proportional  parts)  are  intended  to  facilitate  this 
operation. 

Thus,  for  1834  we  find  D  to  be  24,  and  in  line  with  24,  in  the 
row  P,  we  have  17  in  the  column  having  7  at  the  top.  This  quan- 
tity, 17,  is  to  be  added  to  the  mantissa  of  1834  to  give  the  mantissa 
of  18347. 

(&) 

Ex.  2.  To  find  the  arrangement  corresponding  to  the  mantissa 
26845. 

The  tabular  mantissa  next  below  this  is  26834,  and  the  cor- 
responding arrangement  is  1855. 

The  excess  of  26845  is  11,  and  D  being  23,  we  find  in  line  with 
23,  in  row  P,  that  11  is  in  the  column  having  5  at  the  top.  Then  5 
is  to  be  attached  to  1855,  giving  18555  as  the  arrangement  cor- 
responding to  26845. 

219.  It  must  be  remembered  that  the  mantissa  is 
always  positive,  while  the  characteristic  is  negative  for 
numbers  less  than  1,  zero  for  numbers  from  1  to  10,  and 
positive  for  numbers  above  10. 

To  mark  the  negative  characteristic  the  minus  sign  is 
written  above  the  characteristic  instead  of  before  it. 

Ex.  1.    To  find  the  value  of  (1.8471)*^. 

logl.8471  =  0.26649 

7 


log  (1.8471)7  =  1.86543 
/.  (1.8471)7  =  73.355... 

Ex.  2.  To  find  the  value  of  (18.71)i 

logl8.71  =  1.27207 


264  OF   THE   TABLE   OF   LOGARITHMS. 

Divide  by  5.  log  (18. 71)*  =  0.25441 

/.  (18.71)^  =  1.7964... 

Ex.  3.   To  find  the  value  of  (0.185)^. 

log  (0.185)  =1.26717 
7 


.-.  log  (0.185)7  =  6.87019 

.-.  (0.185)7  =  0.0000074162 ... 

Ex.  4.   To  find  the  value  of  (0.001836)TT. 
log  (0.001836)  =3.26387 
log  (0.001836)5=  14.31935 

=  22  +  8.31935 
.-.  log  (0.001836)TT  ^  2.75631 

and  (0.001836)A  =  0.057056  ... 

Notice  that  to  divide  the  negative  characteristic,  14, 
by  11,  we  make  it  evenly  divisible  by  subtracting  8  from 
it  and  adding  8  to  the  mantissa,  so  as  to  keep  the  whole 
unchanged. 

EXERCISE  XVII.  b. 

(All  the  exercises  here  given  can  be  worked  by  means  of  the 
portions  of  logarithmic  table  given.) 

1.  Find  the  continued  product  of  1.783,  1.791,  and  1.799. 

2.  Find  the  value  of  (18.43  x  18.65  x  1.876  x  5736)  --  (1854 
X  186.6  X  5766). 

3.  Find  the  value  of  (0.1866)^  x  (7.365)^ 

4.  Find  the  value  of  (1.8337)3-3037. 

6.   Given  that  17.80  x  17.977  =  320,  to  find  the  logarithm  of  5. 
6.  To  what  power  must  74  be  raised  to  give  57  ? 


KAPIEHIAN   BASE,   AND   EXPONENTIAL   SERIES.     265 


NAPIERIAN    BASE,   AND    EXPONENTIAL    SERIES. 

220.  Definition.    The  quantity  which  we  have  denoted 
by  e,  and  called  the  Napierian  base  in  Art.  211,  is  the 

limiting  value  of  (1  +  7i)"  as  n  approaches  the  value  zero. 

By  the  Binomial  theorem 

(i  +  ^0"  =  i  +  --^+-Y:2^^+^'nT2T3 — ''^' 

^1  I  1  I  ^(^-^)   I  Ul-^0(l-2r0         _ 
1.2      ^  1.2.3 

=  i  +  i  +  J_+ _!_  +  .. 

1.2^1.2.3         ' 
when   71  =  0. 

By  adding  a  sufficient  number  of  terms,  we  find  for 
the  approximate  value  of  e, 

e  =  2.7182818 ...  to  7  decimals. 

This  peculiar  incommensurable  quantity  is  one  of  the 
most  important  constants  in  mathematics. 

221.  From  the  definition  of  e  in  Art.  220, 

1 
e'  =  the  limit  of  ^1  +  n)"}*,  as  n  approaches  zero. 


266     NAPIERIAN   BASE,    AND   EXPONENTIAL   SERIES. 

But 


i(i+n)V=(i+^)" 


xfx  _^\  X 


1-^-^ 


1.2  1.2.3 

and  when  n  =  0, 

^"  =  '  +  f!  +  S  +  S+l]  +  -     •     •     (^> 

Any  power  of  e  is  got  in  the  form  of  a  series,  by  writ- 
ing the  index  of  the  power  in  the  place  of  x  in  the  series 
for  e'.     Thus, 

-  =  6-^  =  1-1  +  ---  +  --+ - 
e  2!     3!     4! 

and        Ve  =  e*  =  1  +  -  +  -i-  +  -^  +  -^  +  •  •  • 

^  2     2 !  22     3 !  2^     4 !  2* 

222.  Let  a  =  e\  Then  c  =  log^  a ;  or,  denoting,  in 
future,  Napierian  logarithms  by  the  single  italic  I  fol- 
lowed or  not  by  a  point,  c  =  l  -  a. 

And     a'=e«=H-cx  +  ~  +  ^+- 

This  last  series  is  called  the  exponential  series ;  and  it 
expresses  any  power  (x)  of  a  given  number  (a)  in  terms 
of  the  exponent  and  the  Napierian  logarithm  of  the 
number. 


NAPIERIAN  BASE,   AND   EXPONENTIAL   SERIES.     267 
Coi\    Making  x  =  l, 

and  this  series  which  expresses  a  number  (a)  in  terms 
of  its  Napierian  logarithm  is  sometimes  called  the  anti- 
logarithmic  series. 

EXERCISE  XVII.  c. 

1.  Show  that  e  =  limit  of  ( 1  +  -  j    as  n  approaches  oo. 

90         92        94 

2.  Prove  that  (e2  -  1)2  -  8e2  =  |^  + ^  +  ^+ - 

1 

3.  If  X  is  positive,  then  e  >  x'. 

4.  The  series  —  +  — ^— ^ —  + ^'^'^,    ^  +  ...  is  the  ex- 

1.2     1.2.3.4      1.2.3.4.6.6 

pansion  of  (^^e  —  1). 

5.  Find  the  sum  of  e  +  e-i. 

6.  Show  that    I (e''  +  e-«)  =1-^  +  ^-^  +  ... 

2!      4!      6! 

7.  Show  that  — (e'*  -  e-'*)  =x-  —  +  —  -  —  + 

2i  ^  3!      5!      7! 

8.  Expand      ^ 


e*-l 
Assume  the  expansion  to  be      a  +  5ic  +  cx^  +  ... 

Then  a;  =  (a  +  6x  +  cx2  +  ...)  ^x  +  ^  +  ^  4- .-.] 

Distribute  and  equate  coefficients  of  like  powers  of  a,  and  find 
a,  &,  c,  etc. 

9.   Show  that  l^  +  ?i  +  §!.  +  ^+...  =2e. 
1  !      2  !      3  !      4  ! 

10.  Find  the  value  of  i^  +  —  +  —  +  —  4-  - 

11         21314! 


268  LOGARlTHxMIC    SERIES. 


LOGARITHMIC    SERIES. 

223.  In  the  exponential  series  (O,  Art.  222)  the  ]^a- 
pierian  logarithm  of  a  is  the  coefficient  of  linear  x  in  the 
expansion  of  a^  And  as  a;  is  arbitrary  it  follows  from 
the  principle  of  undetermined  coefficients  that  if  we 
expand  a^  in  ascending  powers  of  x,  by  any  means,  the 
coefficient  of  linear  x  in  the  expansion  will  still  be  the 
Napierian  logarithm,  of  a. 


But     a'  =  {l  +  a-iy 

1.2.3  ^         ^ 

by  the  Binomial  Theorem. 

And  picking  out  the  terms  which  form  the  coefficient 
of  linear  x,  we  have 

Z .  a  =  (a  -  1)  -  i(a  -  l)^  +  -i(a  -  1)^-  +  ...       (E) 

which  gives  Z .  a  in  terms  of  the  number  less  by  unity. 
Writing  1  -{-x  for  a, 

Z.(l4-aj)  =  a^-ia^  +  iaj3-^aj^  + (F) 

which  is  the  logarithmic  series. 

224.  Writing  1  for  a;  in  i^ gives  a  series  for  I  -2 ;  but 
this  series  is  so  slowly  convergent  (see  limits  of  a  series. 
Chap.  Xyill.)  as  to  be  of  no  practical  utility  in  compu- 
tations. 

We  transform  the  logarithmic  series  as  follows : 


LOGARITHMIC    SERIES.  269 

Til  (F)  write  —  x  for  x,  and  we  get 

6  •  (  X  —  X)  —  ~~  X  —  -n"  X    "~~"  -5-  3/    —  -T  U/    ~"~  *  *  *  > 

and  subtracting  this  latter  series  from  (F), 

l.l±1^2{x  +  ^x'  +  lx^  +  ...)  ((?) 

1  —  X 

Now   make   x  = ;  then     "*" '  = ,  and  this 

-,  .^.  .  2z  —  l  1  —  x     z  —  1 

reduces  (O)  to 

l.z=l(z-l) 

\2z-l      3    {2z-iy     5    (2z-iy  I       ^     ^ 

This  makes  l-z  depend  upon  I  (z  —  1),  and  a  function 
of  z  which  makes  up  the  difference  between  the  two 
logarithms. 

Ex.  1.  Let  z  =  2.     Then  since  M  =  0, 

^. 2  =  2(1  +  1. -1+1. I+.4 
l3      3    33^5    35^      / 

=  0.69315...  to  five  places. 

Ex.  2.  Let  z  =  6.    Then  since  Z .  4  =  2  ? .  2, 

Z.6  =  2Z.2  +  2/l  +  l.i  +  l.i  +  ...\ 
193    93      5    95  / 

=  1.60944 ...  to  five  places. 
Ex.  3.     Z .  10  =  Z .  2  +  Z .  5  =  2.30259  to  five  places. 

225.  The  series  now  obtained  furnishes  a  practical 
method  for  computing  logarithms  to  the  base  e. 


270  LOGARITHMIC   SERIES. 

Now  let  a'  =  6,  and  take  the  Napierian  logarithms  of 
both  members  of  the  equation ;  then 

ha'  =  xl'a  =  l'b, 
and  X  =  log  J). 

.-.  log.&  =  -^.Z.&. 
Z-a 

Hence is  a  multiplier  which  changes  I  •  b  into  log„5. 

ha 

And  a  being  given,  T  a  is  also  given  and  constant,  being 
the  Napierian  logarithm  of  the  given  base. 

The  multiplier,  {l-a)~\  is  called  the  modulus  of  the 
system  of  logarithms  having  a  as  base. 

The  modulus  for  base  10,  or  (MO)-^  is  0.43429448... 
to  8  decimals. 

Thus  the  Napierian  logarithm  of  any  number  is 
changed  into  the  decimal  logarithm  of  the  same  number 
by  being  multiplied  by  0.43429448  •«. ;  and  the  decimal 
logarithm  of  any  number  is  changed  into  the  Napierian 
logarithm,  by  being  divided  by  0.43429448 ...  or  by  being 
multiplied  by  the  reciprocal  of  this  quantity,  namely, 

2.30258509...,  which  is  Z.IO. 

The  logarithms  of  any  two  systems  are  thus  connected 
by  a  constant  multiplier,  the  modulus  of  one  system 
Avith  respect  to  the  other. 

Napierian  logarithms  are  most  convenient  in  analysis, 
and  decimal  logarithms  in  practical  applications,  and  the 
change  from  one  system  to  the  other  is  easily  effected. 


LOGAKITHMIC   SERIES.  271 

EXERCISE  XVII.  d. 

1.  Find  X  from  the  equation  a(h  +  1)  =  I  •  (a  +  ic). 

2.  Show  that  a;  =  e«-*  =  a^«  =  a^°^«"'. 

1  1  1 

3.  li  y  =  e^-^-",  z=  ei-«-%  then  x=  e^-^-y, 

4.  Prove  that  Z.a-?..x=  ^^^ -^f  ^:^YV-V^^^V-  +  ••• 
6.  Prove  that 

ii{(x  -V)-\{x-  \y  +  K^  - 1)'  -  •••} 

^x^-\    i/x^-i\2    i/x^-iy 
x«       2V   x»   y     3V    X"   y 

l+ae  1-x  2  4  S 

6.  Show  that  ?.{(!  + a:)  2  .  (i  _  x)  2  }  ^z -^  +  ^- +  ^- +  ... 

^^        ^        ^  ^      ^      1-2      3.4     5.6 

7.  Prove  that  the  ratio  log  a :  log  &,  where  a  and  ?>  are  given 
numbers,  is  the  same  for  all  systems. 

8.  The  modulus  which  changes  from  base   10  to  base  3  is 
log  10 :  log  3,  taken  in  any  system. 

9.  Show    that    log,o9  =  1  -  m  fi  -  i  .  —  +-.  —  -  + -^ 

^^  Vio    2   102    3   103    ^    y 

where  m  is  the  modulus  to  base  10. 

10.  Show  that  the  logarithm  of  a  number  cannot  be  developed 
in  terms  of  the  number  itself. 

226.  The  exponential  and  the  logarithmic  series  can 
be  obtained  by  other  methods  besides  the  ones  already 
employed.  We  give  some  of  these  as  examples  and 
exercises. 

Ex.  1.    Assume  a^  —  \^-h^x-\-  \x^  +  h.^x'^  +  ... 

Then  a^=\  +  5i(2x)  +  &2(2^)2  +  &3(2a;)3  +  "•       (1) 


272  LOGARITHMIC    SERIES. 

Equate  coefficients  of  like  powers  of  x  in  the  two  expressions 
for  d^. 

Ex.  2.    In  Ex.  1,  &i  is  indeterminate  ;  what  does  it  mean? 

Ex.  3.    How  do  we  know  that  the  expansion  of  a^  must  begin 
with  1? 

Ex.  4.   If  a  =  e*S  show  from  the  result  of  Ex.  1  that 

Ex.  6.  Assume    a^  =  \ -\- h^x -\-  h.^x'^  +  ...  hnX""  +  ••• 

Then  ay  =  I  -\-  h^y  -\-  h.^y'^  -\ 6„?/"  +  ... 

And  a^  -  av  =  h^{x  -y)^-  h^ipcP'  -  y'^)  +  —  6„(x"  -  ?/«)  +  ...   (3; 
But  w"-  aM  =  ayia^'-y  -  1) 

=  ay{b,{x-y)  +  h^{x  -  ?/)2+  ••.  6„(x  -  yy^- ...}  (4) 

Make  (3)  =  (4)  as  they  are  equivalents;  divide  throughout  by 
X  —  y,  which  is  a  factor  ;  put  y  =  X]  and  equate  coefficients  of  x'^. 

Then  6„^i  =  _k^.6 

Erom  this  relation  obtain  the  coefficients  in  terms  of  5^,  which 
is  indeterminate. 

Ex.  6.   In  Ex.  6,  why  is  it  necessary  to  divide  hjx  —  y  before 
making  y  =  x? 

Ex.  7.   If         e«  =  6,  cc  =  Z .  6. 

But  5  _1  =  5C +  —+-  +  .  .. 

2!      3! 

Assume  x  =  A(b  -  l)-\-B(b  -  1)^  +  C(b  -  ly  + -"    (5) 

In  (5)  put  the  value  of  5  —  1  taken  from  the  preceding  line,  and 
equate  coefficients  of  like  powers  of  x.     Then  find  A,  B,  C,  etc. 
This  gives  L  6  in  terms  of  6  —  1. 


LOGARITHMIC   SERIES.  273 

Ex.  8.  Starting  from  a*  =  6,  proceed  as  in  Ex.  7  to  find  loga  b 
in  terms  of  b'—  1.  What  is  the  meaning  of  the  indeterminate  co- 
efficient in  the  result  ? 

Ex.  9.    Assume  log  (1  +  x)=  ax+  bx'^  +  cx^  +  •.. 
Then  n  being  an  arbitrary  quantity, 

n\og(l+x')=n{ax+bx^  +  cx^  +  ''-)     .    •    .     (6) 
But  n  log  (1  +  x)  =  log  (1  +  x)« 


=  log  (1  +  «(7iX  + "02^2 +...) 
=  a(^C,x  +  »a,x^  +  ...)+  bi^C^x  +  ^a,x^  +  "-y  +  -  .    (7) 
by  taking  the  expression  ^C^x  +  ^C^x^  +  •••  as  a  variable. 

(6)  and  (7)  are  equivalents.  Make  /i  =  0,  and  equate  coefficients 
of  like  powers  of  x. 

The  result  contains  the  indeterminate  a.  What  is  this,  and 
what  is  the  effect  of  making  a  =  l? 

Ex.  10.  In  the  assumption  of  Ex.  8,  how  do  we  know  that  the 
first  term  of  the  expansion  must  contain  x  ? 


CHAPTER  XVIII. 
Of  Series. 

227.  Series  are  too  varied  in  character  to  be  rigidly- 
classified,  but  the  greater  number  of  them  have  a  relation 
to  the  geometric  series,  or  to  the  arithmetic  series,  or  to 
both. 

In  series  related  to  the  geometric,  any  term  is  con- 
nected with  one  or  more  of  the  preceding  terms  by 
constant  multipliers,  or  by  multipliers  which  vary  with 
the  number  of  the  term  in  the  series. 

Thus  in         1  +  3  +  7  +  15  +  31  +  63  +  ... 

63  =  3  X  31  -  2  X  15,  31  =  3  x  15  -  2  x  7,  etc.. 

with  constant  multipliers,  3  and  —  2. 

In  i  +  £4-^  +  ^%... 

1!      2!      3! 

the  wth  term  is  got  from  the  (w  —  l)th  term  by  multiplying  by  -, 
and  the  multiplier  is  a  function  of  the  position  of  the  term  in  the 


228.  The  nth  term  of  a  series  is  such  a  function  of  n 
that  after  some  particular  term,  usually  near  the  begin- 
ning of  the  series,  any  term  is  got  by  writing  the  number 
of  the  term  for  n  in  the  function  of  n. 

Usually,  however,  when  the  series  contains  a  variable, 
X,  in  ascending  powers,  the  absolute  term  is  not  con- 
274 


BECURRING   SERIES.  275 

sidered  in  the  counting,  so  that  the  term  counted  as  the 
nth  is  the  (n  +  l)th  from  the  beginning.  This  usage 
makes  the  nth.  term  contain  a?" ;  and  in  purely  numerical 
series  a  unit  variable  is  frequently  introduced  for  this 
and  other  purposes. 

Thus  (2'»+i  —  l)x"  is  the  nth.  term  of  the  series 

RECURRING    SERIES. 

229.  A  recurring  series  is  generally  the  expanded  form 
of  a  proper  fraction,  and  is  analogous  to  the  circulating 
decimal  in  arithmetic. 

Thus  — —  =:l-\-x  +  x^  +  x^-] — ,  a  geometric  series. 

1  —X 

h±ll =l  +  5a;+13«2  +  29x3  +  61cc*+..., 

1  —  3  X  +  2  ic2 

a  recurring  series  of  the  second  order. 

1j=l^ =l_2a  +  4x2-7x3  +  13a:*-23«5_f-... 

1  +  X-2X2-X3  ' 

a  recurring  series  of  the 'third  order. 

etc.        etc.        etc. 

In  this  relation  the  fraction  which  by  its  expansion 
produces  the  series,  is  called  the  Generating  Function 
(G)  of  the  series ;  and  the  denominator  of  the  fraction 
is  the  Scale  of  Relation  (E). 

When  the  E  is  binomial  and  linear,  the  series  is 
geometric,  and  is  of  the  first  order ;  and  generally,  the 
order  of  the  recurring  series  is  the  same  as  the  dimen- 
sions of  the  K. 


276 


RECITERIKG  SERIES. 


230.  Problem.  G-iven  a  recurring  series  and  its  order, 
to  find  its  E;  and  its  G-. 

Take  the  recurring  series  of  the  2d  order  given  above, 
viz. : 


Assume 


N 


=  l  +  5aj+13a;2+29a^+61a;*+. 


1  -\-  ax -\- bo(^ 

Multiply  by  the  denominator.     Then 

JV^=  1  +  5 
a 


ic  +  13 

aj2  +  29 

0^3  +  61 

5a 

13  a 

29  a 

b 

5b 

13  b 

x^  +  • 


But  N  cannot  be  higher  than  linear  in  x ;  and  there- 
fore all  the  coefficients  after  linear  x  must  vanish. 

That  is, 

13 -}_    5a+      &  =  0; 

29  + 13a  +    5&  =  0; 
61  + 29a +  136  =  0,  etc. 

The  first  two  equations  give  a  =  —  3,  5  =  2,  and  these 
values  satis.fy  the  third  equation. 

This  compatibility  shows  that  the  series  is  of  the 
second  order  and  the  R  is  quadratic. 

If  the  third  equation  were  not  satisfied,  the  E,  would 
be  of  higher  dimensions,  and  the  series  would  be  of  a 
higher  order. 

TheEis  l-3a;  +  2a;2. 

By  putting  —  3  for  a  in  the  terms  up  to  the  linear 
inclusive,  we  get  JV=  1  +  2a;. 

l  +  2a; 


G  = 


l_3a;  +  2ic2 


BECURRING  SERIES.  277 

231.  We  see  from  the  foregoing  article,  that  if  z  be 
the  order  of  the  series,  it  requires  z  terms  to  find  the 
E,  and  z  terms  to  find  N.  Hence  the  number  of 
terms  required  to  determine  completely  a  recurring 
series  of  any  order  is  twice  the  number  of  the  order; 
and  if  z  be  the  order,  2;  —  1  terms  at  most,  counting  from 
the  beginning,  may  not  follow  the  law  of  the  rest  of  the 
series. 

232.  Problem.    To  find  the  7ith  term  of  a  recurring 

series. 

Taking  the  series  of  Art.  230,  its  R  factors  into 
{l  —  x)(l—2x)j  and  going  to  partial  fractions, 

l  +  2x       ^4  3 

l^^x-^2x'~l-2x     1-x 

Then  expanding  these  partials,  we  get  the  equivalent 
geometric  series : 

=  4(l  +  2a;4-2V+...2"a;"+...). 
=  3(1 +  aj  +  a;2  4-...aj« +  ...). 


l-2a; 
3 


And  confining  ourselves  to  the  coefficient  of  a;", 
(2«+2  _  3)aj^  =  the  nth  term. 

This  shows  that  the  terms  of  a  recurring  series  are,  in 
general,  the  algebraic  sums  of  the  corresponding  terms 
of  two  or  more  geometric  series. 

And  in  finding  the  nth  term  we  need  not  write  out  the 
geometric  series ;  for  if 

z ,     -^ -^,     z ,     etc., 

\  —  ax      1  —  f3x      1  —  yx 


278  BECURRING   SERIES, 

be  the  partial  fractions, 

is  the  72th  term. 

233.  Our  ability  to  find  the  nth  term  depends  upon 
our  ability  to  factor  the  E-.  If  the  R  rises  only  to  a 
quadratic,  or  is  separable  into  factors  none  of  which  are 
higher  than  quadratic,  it  is  possible  to  find  the  rith  term ; 
but  when  the  linear  factors  are  irrational,  the  operation 
may  be  laborious. 

We  give  one  example. 

Ex.  The  series    1  -  4  ic  +  19  a^^  -  17  x^  +  265  x* 

has  1  +  4  a;  +  ic2  as  its  R,  and  N  =  1  —  x. 

1  +  4x  +  X2  =:  {1  +  (2  +  ^S)X}{1  +  (2  -  ^S)x], 

Assuming    — ^-^^^ —  = ^ +  -  ^ 


l  +  4a:  +  ic2      i+(2-f^3)x      l-i-(2-y/S)x 
we  obtain         ^  =:  J  (1  +  -y/S),  B=i(l  - ^S). 
Thence  the  nth  term  is 
(-)**  H(2  +  V3)"  +  (2  -  V^)"  +  V[(2  +  V^)"  -  (2  -  V3)"]}aJ«. 

EXERCISE  XVIII.  a. 

1.  Find  the  R  and  the  G  of  1  +  2  x  +  3  cc2  +  4^3  +  ...,  it  being 
of  the  second  order. 

2.  Eind  the  R  and  G  of  1  +  3  x  +  4 x^  +  5  x^  +  •••,  a  recurring 
series  of  the  second  order. 

3.  Find  the  next  two  terms  of  the  series  of  Ex.  2. 

1+  2  X  -  x2 


4.  Develop  the  series  whose  G  is 


(l-x)(l  +  x)2 


DIFFEKENCE   SERIES.  279 

5.  The  terms  1  +  x  —  2  x^  +  3  ic^  are  the  first  four  terms  of  a 
recurring  series  of  the  second  order,  and  also  of  one  of  the  third 
order.    Find  the  G's  of  the  series,  and  the  5th  term  in  each. 

6.  The  first  four  terms  of  a  recurring  series  of  the  third  order 
are  1  —  x  + 2x^  —  2x^.  Find  an  expression  for  the  nth  term,  and 
thence  find  the  99th  term. 

7.  Find  the  nth  term  of  the  series  of  Ex.  4. 

8.  Find  the  nth  term  of  the  series  of  the  second  order, 

1  +  5x+  19x2  +  65x3+  ... 

9.  If  there  be  n  terms  given,  N  may  contain  any  number  of 

terms  from  1  to  -  —  1  if  w  is  even,  and  from  1  to  ^  ~     if  n  is  odd. 
Z  2 

10.  If  n  terms  be  given,  they  may  belong  to  a  series  of  any 
order  from  w  —  1  to  -  +  1  if  w  is  even,  and  from  n  —  1  to  ILtA 
if  n  is  odd.  ^  2 


DIFFERENCE    SERIES. 

234.   These  have  an  alliance  with  arithmetic  series. 
Take  the  series 

2  +  3  +  6  +  12  +  22  +  37  + — 
1st  differences         1  +  3  +  6  +  10  +  15  + 
2d  «  2  +  3  +  4  +  5  + 

3d  "  1  + 1  + 1  + 

4th         "  0  +  0  + 

By  subtracting  each  term  from  the  following,  we  obtain 
a  set  of  series  similar  to  the  first,  but  of  successively 
lower  orders,  called  the  series  of  1st  differences  or  Aj- 
series,  the  series  of  2d  differences  or  Aj-series,  etc. 


280  DIFFERENCE   SERIES. 

In  the  example  given,  the  Ag-series  is  arithmetic,  and 
the  A4-series  vanishes ;  and  for  any  true  difference  series 
some  A-series  is  arithmetic,  and  the  second  one  there- 
after vanishes. 

Thus  in  the  series  of  cubes  1,  8,  27,  64,  125,  etc.,  the 
Ag-series  is  arithmetic.  In  1^,  2^,  3^,  etc.,  the  A4-series 
is  arithmetic,  etc. 

Evidently  if  any  general  relation  exists  between  the 
original  series  and  its  A^-series,  a  similar  relation  must 
exist  between  each  two  consecutive  A-series. 

235.  Let  Uq  +  Ui-\-  U2-\-  •*-  be  a  difference  series  in 
which  the  suffix  serves  the  purpose  of  the  exponent  of  a 
variable. 

Then,  -w-series   t^o  +  ^^i  + '^2  +  ^s  +  ^4 -f- ••• 
Ai-series        Ai      A/     A/'    A^'" 
Ag-series  Ag     Ag'    A2" 

Ag-series  A3    A3' 

A4-series  A^ 

Now,        Ui  =  Uo-{-Ai,  A/  =  Ai4-A^ 

A2'  =  A2  +  ^3?  ^3'  =  A3  +  A4,  etc. 
Again,     U2  =  Ui-\-  A/  =  2^0  +  2  Aj  -[-  Ag. 
.-.  Ai"  =  Ai  +  2A2  +  A3. 
A2"  =  A2  +  2A3  +  A4. 
Again,      ^(3  =  U2-{-  A/'  =  -i^o  +  3  Ai  +  3  Ag  +  A3. 

.-.  A/"  =  Ai  +  3A2  +  3A3  +  A4. 
Again,      u^=zUs  +  Aj'"  =  i^o  +  4  Ai  +  6  Ag  +  4  A3  +  A4. 

And  obviously,  from  the  mode  of  formation  of  the 
terms, 

^„  =  ^0  +  "CiAi  +  "C2A2  +  "C3A3  +  ... 


INTERPOLATION. 


281 


which  is  an  expression  for  the  nth.  term  of  a  difference 
series. 

Ex.   To  find  the  nth  term  of  2  +  3  +  6  +  12  +  22  +  ••• 
u,  =  2,  \  =  1,  A,  =  2,  A,  =  l,  A,  =  0. 

/.  u^  =  2  -\-n  +  n{n  -  1)  +  |  n(w  -  1) (n  -  2) 


INTERPOLATION. 

236.    Take  the  difference  series  whose  first  5  terms  are 

7,  2,  1,  4,  11. 
The  expression  for  the  nth.  term  is 

u^  =  7  -^n(2n  —  T). 
Eegarding  ti  as  a  variable  of  the  function  whose  value 
is  denoted  by  u^y  we  draw,  as  in  the  figure,  the  graph  (G) 
of  the  function,  in  which  7i  takes  the  place  of  x^  and  u^  of  y 


Then 


Oa: 


^  =  7,     lb  =  u,  =  2, 


2c  =  U2  =  l,     3 d  =  %  =  4,  etc. 
And  the   points   a,  6,  c,  d,  etc.,  represent,  by  their 


4 


V 


ordinates,  Oa,  lb,  etc.,  the 
terms  of  the  series. 

Hence  we  may  define  a 
series  as  a  set  of  point  values 
of  a  function  of  a  variable, 
corresponding  to  equidis- 
tant values  of  the  variable; 
the  equidistant  values  being 
generally  regarded  in  all 
series  as  the  consecutive  in- 
tegers from  zero  upwards,  or  the  numbers  of  the  succes- 
sive term. 


Xm' 


Li 


I     1    m    2     ^    3 


282  INTERPOLATION. 

Now  n  =  I  gives  the  point  value  W  corresponding  to 
the  middle  point  of  01;  w  =  f  gives  mm'  corresponding 
to  the  middle  point  of  1  2 ;  etc.  And  the  points  0,  Z, 
1,  m,  2,  etc.,  being  equidistant,  we  have  a,  Z',  6,  m\  c,  etc., 
as  terms  of  a  new  series,  such  that  every  alternate  term, 
counting  from  the  first,  belongs  to  the  original  series. 

We  are  then  said  to  have  interpolated  single  mean 
terms  in  the  original  series. 

Our  unit  on  the  ic-axis  being  arbitrary,  we  may  make 
0 1  the  unit  by  writing  ^  n  for  n  in  the  function,  and  leav- 
ing u^  unchanged. 

This  gives  ^^^  =  7  4.  ^  (n  -  7) 

for  the  new  series ;  and  the  series  itself  is 

7,  4,  2,  1,  1,  2,  4,  7,  11 ... 

So  that  IV  —  4,  mm^  =  1,  nn'  =  2,  etc. 

In  a  similar  manner  by  writing  -  for  n  in  the  nth 

o 

term  of  the  original  series,  we  obtain  the  nth  term  of  a 
series  in  which  two  mean  terms  are  interpolated  between 
each  two  consecutive  terms  of  the  given  series,  etc. 

In  like  manner,  if  any  real  value  whatever  be  given  to 
n,  the  resulting  value  of  u^  is  the  ordinate  corresponding 
to  that  particular  value  of  n. 

237.  The  least  consideration  will  show  that  interpola- 
tions can  be  made  accurately  whenever  the  nth  term  can 
be  accurately  expressed,  and  that  the  last  condition  is 
satisfied  for  any  series  in  which  an  order  of  differences 
becomps  zero.  Also,  that  if  no  order  of  differences  is 
zero,  the  ?ith  term  can  be  expressed  only  approximately, 


INTERPOLATION.  283 

and  the  interpolated  terms  will  be  only  approximately 
eorrect. 

As  an  illustration  of  the  latter  statement  consider  a 
t  ible,  such  as  that  of  logarithms,  for  example. 

The  logarithmic  series  is  a  function  of  a  variable  n, 
and  the  tabulated  logarithms  are  the  point  values  of  this 
function  corresponding  to  consecutive  equidistant  values 
of  the  variable,  as  30,  31,  32,  etc.,  say. 

These  logarithms  do  not  form  a  proper  difference  series, 
and  the  rith  term  cannot  be  exactly  expressed. 

Thus 


log  30  =  1.47712 


Ai 


1424 
log  31  =  1.49136         ^  -  45 

1379 
log  32  =  1.50515  -43 


log  33  =  1,51851 


1336 


But  A2  is  small  as  compared  with  A^,  and  nearly  con- 
stant, so  that 

u,  =  1.47712  +  1424  n  -  -V-n(n  -  1) 

is  approximately  true  for  small  values  of  n,  as  from  0  to 
1 ;  i.e.  the  result  will  be  practically  correct  for  the  loga- 
rithm of  any  number  lying  between  30  and  31. 

Thus  log  30.3  =  1.47712  +  -j-% .  1424  +  S^-  •  y'lr  •  tV 

=  1.47712  +427  +  5 

=  1.41844, 

which  is  true  to  the  last  figure. 

This  example  shows  that  in  a  case  like  the  present 
proportional  parts  are  not  always  sufficient. 


284  SUMMATION   OF    SERIES. 


EXERCISE  XVIII.  b. 


1.  Find  the  orders  of  differences  of  the  difference  series,  50,  52, 
50,  45,  38,  30,  etc. 

2.  Find  the  expression  for  the  nth  term  of  Ex.  1. 

3.  Find  the  nth.  term  of  the  difference  series  of  which  the  first 
four  terms  are  1  +  7  +  11  +  13. 

4.  Find  the  nth  term  of  li,  2,  3,  4|,  8  ... 

5.  Interpolate  mean  terms  in  the  series  4,  1,  2,  7. 

6.  If  a,  5,  c  be  three  consecutive  terms  of  a  difference  series, 
and  m  and  n  be  mean  terms  between  a,  b  and  6,  c,  respectively, 
show  that  m  =  ^(3  a  +  6  6  —  c),  and  n  =  J(3  c  +  6h  —  a). 

7.  If  in  Ex.  6,  m,  n  be  two  interpolated  mean  terms  between 
a  and  6,  and  j?,  q  be  two  between  b  and  c,  show  that,  upon  the 
supposition  that  A3  =  0,  m  =  i(5  a+56  —  c),  n  =  |(2a  +  8&  —  c), 
p  =z  1  (2  c  +  8  &  -  a),  and  g'  =  i(5 c  +  5  6  -  a). 

8.  The  expectation  of  life  at  10  years  of  age  is  48.8,  at  20  it  is 
41.5,  at  30  it  is  34.3,  and  at  40  it  is  27.6,    What  is  it  at  15 ?  at  25  ? 

9.  At  9  o'clock  the  distance  of  a  star  from  the  moon  is  42',  at 
10  it  is  19',  and  at  11  it  is  —  3'.  How  were  they  situated  at 
10  h.  52  m.  ? 

10.  Given  sin  240  =0.40674,  sin  25°=  0.42262,  sin  26°=  0.43837, 
sin 27°=  0.45399;  find  sin 24°  25'. 


SUMMATION    OF    SERIES. 

238.  The  Sum  of  a  Series  is  a  somewhat  indefinite 
expression,  as  the  following  statements  will  show. 

(1)  If  the  series  be  numerical,  the  sum  of  its  first  n 
terms  is  intelligible,  whether  a  general  expression  for 
such  a  sum  can  be  found  or  not. 

Thus  the  sum  of  n  terms  of  the  series  1-1-2  +  3+  •••  is  Jw(ri+1), 
for  all  values  of  n  ;  and  the  sum  of  any  given  number  of  terms  of 


SUMMATION   OF   SERIES.  285 

the  series  l  +  i  +  i  +  i+---  may  be  found,  although  no  general 
expression  for  the  sum  of  n  terms  has  ever  been  obtained. 

(2)  In  many  numerical  series  the  sum  of  the  series  to 
infinity  may  be  given  as  a  finite  expression ;  but  as  we 
cannot  properly  speak  of  summing  an  infinite  number  of 
terms,  this  expression  is  more  correctly  spoken  of  as  the 
limit  of  the  series,  i.e.  the  value  towards  which  the  sum 
of  the  terms  approaches,  as  more  and  more  of  the  terms 
are  included  in  the  summation,  and  to  which  the  sum 
may  be  made  to  approach  as  near  as  we  please. 

Thus  2  is  the  limit  of  1  +  J  +  i  +  J  H ad  inf. 

(3)  If  a  series  contains  a  variable  in  ascending  or 
descending  powers,  what  is  called  the  sum  is  in  reality 
the  Generating  Function  of  the  series. 

Thus  e""  is  the  G.  of  1  +  a;  +  —  +  .  • .  ad  inf.,  and  can- 

Z  I 

not  be  spoken  of  as  the  sum  of  the  series  without  extend- 
ing the  meaning  of  the  word  sum  quite  beyond  that 
usually  given  to  it. 

Similarly, —  is  frequently  spoken  of  as  the  sum  of 

1  +  ^  +  ^  •  •  •  4-  ^'"~^  because,  when  developed  by  division, 
it  produces  the  series.  But  it  is  evidently  a  generating 
function  rather  than  a  sum. 

Thus  in  reference  to  series,  the  word  sum  applies 
properly  to  a  finite  number  of  terms  of  a  numerical  series. 

The  word  limit  applies  to  an  infinite  numerical  series ; 
and  the  term  generating  function  to  a  series,  finite  or 
infinite,  containing  ascending  or  descending  powers  of  a 
variable. 

We  shall  not,  however,  always  apply  these  distinctions 
rigidly. 


286 


SUMMATION   OF   SERIES. 


SERIES    TO    n    TERMS. 

I.     Generating  Functions. 

239.   As  a  particular  case  take  the  recurring  series  of 
the  second  order ; 


l  +  x 


is  the  G-  of 


l-2x  +  x^ 
Assume 


l-2x-{-x' 
Then  iV:== 


(2n+l)aj'»+...adinf. 


1+3 

-2 


x-i-5 
-6 
+1 


aj2_^...(2n-fl) 

-2(2w-l) 

(2ri-3) 


--2(2n+l) 
(2n-l) 


.n+l 


+(2n+l)a;'*+2 


*'*      """  1-2X  +  X' 

is  the  generating  function  required;  and  this  fraction, 
by  division,  gives  the  series  to  n  terms,  and  no  more. 

The  variable  x  may  take  any  value  except  1  (as  for 
this  value  the  G„  becomes  indeterminate),  and  the  G» 
becomes  the  sum  of  n  terms  of  a  numerical  series. 

Thus,  putting  a?  =  2,  we  have 

S,=  3  +  (2n-l)2^-^\ 
as  the  sum  of  the  first  n  terms  of  the  series 

1  4-  6  +  20  -f  56  +... (2  n  +  l)2^ 


SUMMATION   OF   SERIES. 


287 


Similarly,  for  a;  =  ^  we  get   ~ ^^ — — ^    as  the 

sum  of  n  terms  of  the  series 

240.   Now  take  the  general  case,  and  let 


l+px-{-  qx^ 


tto  +  ^1^  +  •  •  •  €inX\ 


■^n  =  «0  +  «1 

pao 


x-\ a„ 

PCtn-l 


a;"+^.. 


+ga" 


v.n+2 


And  from  the  property  of  the  R,  that  every  column 
with  three  terms  is  zero, 

^n  =  «o  +  {cti  -i-pcto)^  +  lpc('n  +  Q^tn-i  +  got„a;Ji»"+\ 
Also  a„+i  +  pa„  +  g«n_i  =  0. 

.-.  iV;  =  ao  +  (cti  +i>ao)a;  —  (a„^i  —  ga„a;)aj"+\ 
And  the  required  G  is 

1  -j-  px  +  ga;^ 

In  a  similar  manner  the  G„  can  be  found  for  a  recur- 
ring series  of  any  order. 

241.   To  find  the  sum  of  n  terms  of  the  series  whose 

nth  term  is   • 

n{n-{-l){n-\-2) 

This  series  is  allied  to  a  recurring  series,  but  the  scale 
of  relation  is  not  of  finite  dimensions. 


288 
We  have 


SUMMATION   OF   SERIES. 


n{n  +  l)(n  +  2)      2n      2{n^V)      2(n  +  2) 


And  by  putting  n  =  1,  2,  3,  etc.,  we  express  the  given 
series  as  the  sum  of  three  series,  viz. : 


^r. 


^  +  2-  +  l+' 


2 
2' 


2_ 
3 

4- 


1 

n 
2 

n 


71  +  1 
1 


+  - 


=  '}-■ 

2(2 


2      (^  +  l)(n  +  2)) 
Cor.    If  n  =  00,  we  obtain  as  the  limit  of  the  series  to  oo, 

s=\. 

Series  of  the  form  of  the  foregoing  can  always  be 
summed  when  the  numerator  of  the  Tith  term  is  constant, 
and  the  denominator  has  its  factors  of  the  form 

(n  + A:)(^  +  2^0(^  +  3^0•••  etc. 


II.     Difference  Series. 

242.  Let  Uq  +  xi^  -f  ^2  +  •  •  •  u^  be  a  given  difference 
series,  and  let  U'o  +  t/i  +  t/'o  +  •••  be  the  series  of  which 
^0  +  '^1  +  "^^2  +  *••  is  the  first  series  of  differences. 

Then  Uo  =  0,  t/j  =  Wq?  t/g  =  n^  +  ^^u  C/3  =  i^o  +  'ih  +  Wg, 
and  generally   JJ^  =  '?^o  +  ^^1  •••  +  ^n-i  =  ^n-x- 

But  the  7ith  term  of  the  C/'-series  is  given  by 

U"^  =  To  +  "Oii^o  +  "^2^1  +  ^C^^2  +  •  •  •  (Art.  235) 


SUMMATIOI^  OF   SERIES.  289 

and  •.•   Uo  =  0, 

Sn-i  =  nuo  +  "O^Ai  +  "(73A2  +  - 

which  gives  the  sum  of  ?i  —  1  terms  counting  from  the 
2d  term,  or  of  n  terms  counting  from  the  1st. 

Hence  for  the  sum  of  n  terms  from  the  beginning  of 
the  series, 

^n  =  nUo  +  "C2A,  +  "(73A2  +  •  •  • 

Ex.  The  sum  of  the  5th  powers  of  the  first  n  natural  numbers  is  — 
n  +  y  n(n  -  1)  +  30  n(n  -  1)  (?i  -  2)  +  -^^  w(w  -  1)  (n  -  2)  {n  -  3) 
+  Sn(n  -  IXn  -  2)(n  -  S)(n  -  4) 
+  ^(n  -  l)(n  -  2)(n  -  3)(n  -  4)(n  -  6)  ; 

which  reduces  to 

j\n\n  +  ly  (2n^  +  2n  -  1). 

LIMIT    OF    A    SERIES. 

243.  The  limit  of  a  series  is  either  finite  or  infinite. 
When  the  limit  is  finite,  it  is  often  called  the  sum  of  the 
series  to  infinity,  and  the  series  is  said  to  be  convergent. 
In  general,  series  which  are  not  convergent  are  classed 
together  as  divergent,  and  cannot  be  said  to  have  any  sum. 

The  limit  of  a  converging  series  may  be  rational,  or  in- 
commensurable ;  but  in  either  case  the  rational  value, 
or  a  sufficiently  close  approximation  to  the  incommen- 
surable, may  be  employed  in  place  of  the  series  in  com- 
putations. 

Such  is  the  case  with  logarithms,  with  e,  with  trigono- 
metrical functions,  etc. 

To  know  whether  a  series  has  a  sum  or  not,  we  must 
determine  whether  it  is  convergent  or  not. 


290  CONVERGENCY   OF   SERIES. 


CONVERGENCY    OP    SERIES. 

When  a  series  contains  a  variable,  its  convergency 
or  divergency  is  usually  dependent  upon  the  numerical 
value  assumed  by  the  variable. 

Thus  X  being  positive,  the  series  1  +  ^  +  ^^  +  •••  is 
convergent  only  when  cc  <  1. 

Some  series  of  this  kind,  however,  and  especially  such 
as  have  increasing  factorials  in  the  denominators  of  their 
terms,  are  convergent  for  all  numerical  values  of  the 
variable. 

244.    It  is  shown  under  geometric  series  that  the  series 

1  -}-  X  -{-  x^  -\- a^  -\-  *  * '  ad  inf.  has  as  its  limit  when 

1—  X 

X  is  positive  and  less  than  1,  and  hence  that  under  these 

conditions  the  series  is  convergent. 

Now  let   Uq  -\- Ui -{- U2 -}- Us -\-  •  •  •   be  an  infinite   series. 

Then 

^  (   .         Ui        Uo    Ui        Uo    Uo    Ua  ') 

;S=^0il  +-  +  -•- +  -•-•-  +  •••    [' 

(  Uq  Ui     Uq  U2     Ui     Uq  ) 

And  if  each  ratio  —,  —,  — ?  etc.,  be  <  x, 

0    ^1         2 

S<Uo\l+x  +  x'-{-a^+'''l', 
which  is  convergent  ii  x<l. 

Therefore  the  series  Uq  +  Wi  +  1^2  +  •  •  •  is  convergent 
if,  after  some  finite  term,  ^^^^  is  less  than  a  quantity 
which  is  less  than  1  for  all  values  of  n. 


CONVERGENCY   OF   SERIES.  291 


Ex.  1.  The  Binomial  series 

1  +  „x  +  ?L(»^Liix-^  +  ... n(n-l)-(n-r+l)^, ^  ... 
2  !  r ! 

where  n  is  negative  or  fractional,  is  infinite  in  extent. 
To  show  that  it  is  convergent  if  a:  be  <  1. 

Un±i  ^  n(n-  l)»>-(yt-r)  ^^+i r\ 1^ 

Un  (r  +  1)!  n{n  —  1)'"  (n  — r  +  1)  x^ 


\l-hr     1  +  rJ 


But      ^     =  0,  and  — - —  =  1,  when  r  =  oo.    And  the  whole  is 
1 +r  1 +r 

<  1  if  a:  <  1,  which  proves  that  the  series  is  convergent  if  x  <  1. 

Ex.  2.  The  series  1+-^  +  ^  +  ^+...  is  convergent  for  all 
numerical  values  of  ic.  ... 

^^^^^  = — ^  ;  and  for  all  finite  values  of  x,  this  is  <  1,  when  n 
Un       7^  +  1 
is  great  enough,  and  is  zero  when  n  =  ao. 

245.  The  series  Wo  —  Wi  +  ^2  —  '^3  H —  •  •  •?  with  alternat- 
ing signs,  is  convergent  when  each  term  is  greater  than 
the  following  one. 

For  S  =  Uo  —  (ui  — U2)  —  {uq  — u^)— "' 

And  as  Ui  >  Wg,  u^  >  u^,  etc.,  every  bracket  is  positive, 
and 

Again,      S  =  {uo  —  u{)  +  {U2  —  ^^3)  +  {u^  —  %)  H 

And  every  bracket  being  positive,  S  >Uo  —  Ui, 
.',  S  lies  between  Uq  and  Uq  —  w^,  and  is  finite. 

Ex.    The  logarithmic  series 

x-ix'^  +  ix^-lx^-h 

is  convergent  when  a;  <  1. 

For  if  a;  be  <  1,  the  condition  is  evidently  satisfied. 


292  CONVERGENCY   OF   SERIES. 

246.  When  the  sum  of  a  few  of  the  first  terms  of  a 
series  is  a  close  approximation  to  its  limit,  the  series  is 
rapidly  convergent. 

If  the  ratio  u^^^ :  u^  approximates  to  zero  as  n  ap- 
proaches 00,  the  series  is  rapidly  convergent.  But  even 
when  this  ratio  approaches  1  as  ii  approaches  oo,  we  are 
not  justified  in  saying  that  the  series  is  not  convergent, 
as  it  may  even  then  be  slowly  convergent,  and  may  require 
further  examination. 

247.  Theorem.  The  series  1 +  ^  +  ^  +  — +  •••  —  +  ••• 

IS  convergent  it  p  >  1. 

Separate  the  terms  of  the  given  series,  after  the  first, 
into  groups  of  2,  2^,  2^,  etc.,  terms ;  then, 

l>i4.i.  £>lj_i4.l4.i.  I>i4....etc 
.*.  US  be  the  limit  of  the  given  series, 


I.e. 


<^+2fe  +  (2^J+(2^J+' 


But  this  latter  series  is  convergent  if  — —  <  1 ;  that  is, 
ifi>>l  ^'~' 

•*•  ^  +  2^  +  3^  +  ^  +  •••  ^^  convergent  if  p  >  1. 

248.    The  series  Wq  +  Wi  +  ^2  •  •  *  +  ^^n  +  •  •  •  is  convergent 
if  u^  =  — ,  and  _p  >  1. 


CONVEEGENCY   OF   SERIES.  293 

Then,  J^  =  i^±l)!=A+lY 

=  l+£  +  K£^  +  4  +  ... 

Every  term  of  the  right-hand  member  except  the 
first  vanishes  when  n  =  oo.     Hence  the  series 

^o  +  Wi  +  WgH u^^ 

is  convergent  if  the   function  n  \  — ^ —  1  f-  >  1,   when 
n  =  00. 

Ex.   To  examine  the  series  whose  nth  term  is  — i-^. 

1  +  ^3 

t^«+i^l+(r^  +  l)  .l  +  n3^  n^+2n3  +  n4-2  ^  .^  ^henn  =  oo 
Un       1+  (n  +  1)^     1  +  n      w*  +  4  n3  +  ...  +  2        ' 

and  this  test  is  not  sufficient. 

Again,     n{^-\\=     2/i^  +  6n3...     ^  2,  when  n  =.  oo. 
l  Un^\        y      n*  +  2  n^  +  n  +  2 

.'.  the  series  is  convergent. 

The  tests  here  given  are  the  most  useful  tests  of 
convergency.  Eeaders  who  wish  to  make  themselves 
acquainted  with  other  tests  will  find  such  in  larger 
Algebras,  where  special  attention  is  given  to  the  subject, 
or  better,  in  works  on  Finite  Differences. 

249.  When  the  sum  of  n  terms  of  a  series  can  be 
found,  and  is  of  such  a  form  as  to  become  finite  when  n 


294  CONVERGENCY   OF    SERIES. 

is  infinite,  the  series  is  convergent,  and  the  finite  expres- 
sion found  by  making  n  =  oo  is  the  limit  of  the  series. 
This  is  quite  self-evident. 

Ex.    The  sum  of  n  terms  of 1 1 f-  .••  is 


3.4     4-5     5.6  3(n  +  3)' 

and  when  w  =  qo,  the  limit  of  the  series  is  i. 


EXERCISE  XVIII.  c. 

1.  Find  the  G„  of  1  +  2  ic  +  3a;2  +  ...  (n  +  1)  x^. 

2      3       4 

2.  Sum  n  terms  of  the  series  1  -{ 1-^  +  —  +  ••• 

3.  Find  the  Gn  of  1  +  3 x  +  6 0^2  +  ...  ^  (w)  (n  +  1) x^-K 

4.  Sum  n  terms  of  the  series  1  +  2.2  +  3   2^  +  4  .  23  +  ... 

5.  Find  the  G„  of  1  -  2  ic  +  3^2  -  40:"^  + 

6.  Sum  n  terms  of  the  series  1  —  2  +  3  —  4+5  —  6  +  «» 

7.  Sum  n  terms  of  -— -  +  — —  +  — —  +  — 

1.2  2 • o      0.4 

8.  Sum  n  terms  of  — -  +  -—-  +  — —  H — 

1.3  3.5     5.7 

9.  Sum  n  terms  of 1 h 1 

1.4  2.5     3.0 

10.  Show  that  Sn  | ^^ zA  =  sJ-^] ^,  where 

l(n+l)(w  +  2)i  \n  +  iy     n  +  2 

Sn  is  the  sum  of  n  terms  from  the  beginning. 

11.  Find  the  limit  of  i-  (I2  +  22  +  32  +  ...  ^2),  when  w  =  00. 

12.  Find  the  limit  of  i  (l^  +  2^  +  3^  +  ...  w^),  when  n  =  cc. 


CONVERGENCY  OP   SERIES.  295 


13.  Find  the  limit  of  -J-  +  -i-  +  -1-  + ...  ad  inf. 

1.4      2.5      3-6 

14.  Is  the  series  1  +  -  +  llll^  +  1-S.6x^  convergent ? 

2         2.4  2.4-6  ^ 

15.  Under  what  condition  is  1  +  2'^  +  S^x^  +  4^^^  -f  •••  conver- 
gent ? 

16.  Show  that  x  —  \x^  +  ^x^  —  ^-x"^  +  ...  is  convergent  if  x  <  1; 
thence  find  to  four  decimals  the  approximation  to  the  value  of 

l5      3.58      5.55  /         1239     3(239)8  J 

17.  Find  the  limit  of  the  series  in  Ex.  2. 

18.  Find  the  limit  of  1  -  -  +  —  -  -^  +  — 


CHAPTER   XIX. 

Determinants. 

250.  Two  figures,  as  3  and  4,  are  in  order  when  the 
less  precedes  the  greater,  and  they  form  an  inversion 
when  the  greater  precedes  the  less. 

Thus  12  3  4  are  in  order ;  1  3  2  4  has  1  inversion,  3  2  ; 
14  2  3  has  2  inversions,  4  2  and  4  3 ;  and  4  13  2  has  4  inver- 
sions, 4  1,  4  2,  4  3,  and  3  2. 

For  illustration,  take  any  arrangement  of  figures, 

3  14  2  5  6  8. 

This  contains  3  inversions.  Interchange  any  two  con- 
secutive figures,  as  2  and  5;  the  arrangement  becomes 

3  14  5  2  6  8,  and  contains  4  inversions.     Or  interchange 

4  and  2,  and  the  new  arrangement  has  2  inversions. 
Thus  the  interchange  of  any  two  consecutive  figures 

increases  or  decreases  the  number  of  inversions  by  one. 

This  is  readily  seen  to  be  always  the  case;  for  if  the 
figures  be  in  order  before  the  interchange,  they  form  an 
inversion  afterwards,  and  vice  versa,  while  their  relations 
to  the  figures  which  precede  or  follow  them  are  unchanged. 

251.  Starting  with  any  arrangement,  as 

3  14  2  5  6  8, 

let  us  interchange  two  figures  which  are  not  consecutive, 
as  1  and  6.      To  do  this,  we  must  move  1  through  4 
296 


DETERMINANTS.  297 

places  to  the  right,  and  then  move  6  through  3  places 
to  the  left ;  or  we  must  move  1  through  3  places  to  the 
right,  and  6  through  4  places  to  the  left.  In  either  case 
we  make  7  consecutive  changes  in  all;  and  the  num- 
ber of  inversions  is  thus  increased  or  decreased  by  an 
odd  number.  The  new  arrangement,  3  6  4  2  5  18,  has 
10  inversions,  or  7  more  than  the  original. 

Similarly,  if  the  orders  of  any  two  figures  be  denoted 
by  m  and  7i,  where  m<7i,  to  interchange  m  and  n  requires 
us  to  move  m  through  n  —  m  places  to  the  right,  and  then 
to  move  n  through  n  —  m  —  1  places  to  the  left ;  or,  to 
make  2  (71  —  m)  —  1  consecutive  interchanges  in  all.  And 
this  being  an  odd  number  gives  the  important 

Theorem.  —  To  interchange  any  two  numbers  in  an 
arrangement  ^increases  or  decreases  the  number,  of  inver- 
sions by  an  odd  number, 

252.  Consider  the  four-dimensional  term  «!  62  ^3  ^4; 
composed  of  four  letters  with  attached  suffixes,  and  in 
which  both  the  letters  and  suffixes  are  in  order. 

Keeping  the  letters  in  order,  let  us  permute  the  four 
suffixes  in  every  possible  way,  as  aj  63  Cg  c?4,  ag  &i  c^  d^,  etc. 
As  we  can  permute  the  four  suffixes  in  ^P^  or  24  ways, 
we  shall  have  24  terms  in  all,  of  which  no  two  have  the 
same  suffixes  attached  to  the  same  letters,  or  are  wholly 
alike. 

The  term  ai  b^  C3  d^,  being  the  one  from  which  the  others 
are  derived,  is  called  the  principal  or  leading  term ;  but 
as  the  whole  set  of  terms  may  be  derived  from  any  one 
of  them,  any  term  may  be  taken  as  a  principal  term. 

Let  us  take  as  our  principal  term  that  one  having  no 
inversions,  and  calling  this  positive,  let  us  agree  that  in 


298  DETERMINANTS. 

forming  the  other  terms  every  inversion  is  to  be  accom- 
panied by  a  change  of  sign.  Then  a  term  with  one  in- 
version in  its  suffixes,  as  aj  b^  Cg  d^,  is  negative  ;  a  term 
with  two  inversions,  as  ai  b^  c^  d^y  is  positive ;  and  gener- 
ally, a  term  is  -\-  or  —  according  as  its  suffixes  contain  an 
even  or  an  odd  number  of  inversions. 

These  considerations  apply  to  a  leading  term  of  any 
number  of  letters,  and  its  derived  terms. 

253.  A  Determinant  is  the  algebraic  sum  of  all  the 
terms  that  can  be  derived  from  a  leading  term,  by  per- 
muting the  suffixes  without  changing  the  letters,  each 
term  being  taken  with  its  proper  sign. 

A  letter  with  its  attached  suffix  is  an  element  of  the 
determinant,  and  as  each  letter  takes  in  turn  each  suffix, 
if  there  are  n  letters,  there  are  also  n  suffixes  and  n^ 
elements. 

A  determinant  with  n  letters  and  n  suffixes  is  of  the 
nth  order,  and  contains  n !  terms. 

The  determinant  of  the  second  order  is  afi^  —  ^2^1?  ^^^ 
of  the  third  order  it  is 

254.  A  letter  with  its  attached  suffix,  standing  as  an 
element  of  a  determinant,  is  symbolic,  and  may  be 
replaced  by  any  quantitative  symbol  whatever. 

But  it  is  only  through  this  symbolic  and  symmetrical 
notation  that  we  are  enabled  to  discuss  with  any  facility 
the  general  properties  of  determinants.  Moreover,  owing 
to  their  unwieldiness  when  written  at  length,  it  becomes 
necessary  to  employ  some  symbolic  or  contracted  form 
for  the  whole  expression. 

The  symbols  :S±ai&2C3---   and  |aAc3---|,  where  the 


DETERMINANTS. 


299 


«!    61 

Ci 

^1 

a2  62 

C2 

d2 

ag     63 

C3 

ds 

a^  64 

C4 

d. 

leading  term  is  written  after  2  ±  or  between  straight-line 
brackets,  are  both  employed.      But   the   working  form 
known  as  a  matrix  is  made  by  writing  the  elements  in  a 
square  between   parallel  vertical  lines,  in 
such  a  manner  that  all  the  same  letters 
stand  in  the  same  column,  and  all  the  same 
suffixes  are  situated  in  the  same  row. 

The   determinant   of    the   4th   order   is 
written  as  a  matrix  in  the  margin. 

255.  The  diagonal  of  the  matrix  from  the  upper  left- 
hand  corner  to  the  lower  right-hand  corner,  namely, 
tti  62  C3  d^,  is  the  principal  or  leading  diagonal,  as  giving 
the  principal  term  of  the  determinant. 

The  sign  of  the  matrix  as  a  whole  depends  upon  that 
of  its  principal  diagonal. 

The  matrix  being  a  symbolic  form  for  a  determinant 
must  be  capable  of  being  expanded  so  as  to  give  the 
determinant,  and  with  a  matrix  having  symbolic  elements 
this  expansion  can  be  effected  by  permuting  the  suffixes 
of  the  leading  diagonal  according  to  the  definition  of 
Art.  253.  Hence  two  matrices  containing  the  same  sym- 
bolic elements  can  differ  only  in  sign,  and  the  signs  of 
two  such  matrices  will  be  the  same  or  opposite  according 
as  the  number  of  inversions  in  their  principal  diagonals 
differ  by  an  even  or  by  an  odd  number,  Art.  2b2. 


256.    Consider  the  matrices 


0) 


a^  hi  Cj  di 

a2  &2  ^2  ^2 

«3  ^3  ^3  ^3 

,    (2) 

a^  64  0*4  d^ 

ci|  0^-2  (X3  a^ 
61  &2  h  h 

Ci    C2    C3   C4 

di  ^2  r/g  ^4 


«!  hi  Cx  c?i 

,    (3) 

«3  ^3  C3  ^3 

^2  62  C2  do 

a^  64  C4  d^ 

300  DETERMINANTS. 

(1)  is  the  standard  matrix  of  the  4th  order,  the  suf- 
fixes being  in  order  in  the  columns,  and  the  letters  being 
in  order  in  the  rows.     Its  principal  diagonal  is  aj  h^  c^  d^. 

(2)  differs  from  (1)  in  having  the  rows  of  (1)  for  its 
columns  and  the  columns  of  (1)  for  its  rows,  the  letters 
and  suffixes  still  being  in  order.  The  principal  diagonal 
of  (2)  is  «!  &2  ^3  ^4?  ^11  d.  being  the  same  as  that  of  (1),  the 
expansions  give  the  same  determinant. 

Therefore,  a  matrix  is  not  changed  in  value  by  chang- 
ing its  rows  to  columns  and  its  columns  to  rows,  provided 
the  letters  and  suffixes  maintain  the  same  order.  Hence 
whatever  is  true  for  a  matrix  with  respect  to  its  columns,  is 
true  also  ivith  respect  to  its  rows,  and  vice  versa,. 

(3)  differs  from  (1)  in  having  its  2d  and  3d  rows  in- 
terchanged. This  introduced  one  inversion  into  its  prin- 
cipal diagonal,  and  hence  changes  the  sign  of  the  matrix. 
And  it  is  readily  seen  from  the  principles  of  Art.  252, 
that  the  interchange  of  any  two  rows,  or  of  any  two 
columns  in  a  matrix  changes  the  sign  of  the  matrix, 
since  it  increases  or  decreases  the  number  of  inversions 
in  the  principal  diagonal  by  an  odd  number. 

257.  Theorem.  If  tivo  columns  or  two  rows  of  a  matrix 
he  identical,  the  value  of  the  matrix  is  zero. 

For,  by  interchanging  the  identical  columns  or  the 
identical  rows,  the  matrix  changes  sign.  But  the  columns 
being  identical  leaves  the  matrix  unchanged.  The  only 
quantity  or  expression  which  remains  unchanged  when 
you  change  its  sign  is  zero.  Hence  the  value  of  the 
determinant  is  zero. 

Thus  the  matrix  with  two  a-columns,  as  in  the  mar- 
gin, expands  into  a^a.fi.^  —  aia.^b.^-\- a./i.^b^  —  a^a^h^-^ a.^ai^b.2 
—  a-/(:J)^,  which  is  identically  zero. 


a. 

a,  b, 

«2 

a,  b. 

a. 

«j  h 

DETERMINANTS. 


301 


258.  Expansion  of  the  Matrix.  Let  us  take  a  matrix 
of  the  third  order  to  begin  with. 

As  every  term  in  the  determinant  contains  each  letter 
once  and  each  suffix  once,  the  terms  which  contain  a^  can- 
not contain  any  other  a  or  any  other  letter  with  suffix  1. 
Hence  the  coefficient  of  aj  is  the  sum  of  all  the  terms 
that  can  be  made  from  the  remaining  letters  and  the 
remaining  suffixes.     But  this,  for  a  determinant  of  the 

3d  order,  is  the  expression  denoted  by  the  matrix 


Similarly,  the  coefficient  of  ag  is 
ho  Co 


h  C3 


&2  ^2 


&3C3 
and  of  ttg  it  is 


Hence  the  expansion  takes  the  form 


±0^1 


&2    C2 

±^2 

5i  c/ 

±a. 

h  Ci 

h  cs 

^3  C3 

&2    ^2 

But  taking  the  principal  diagonals,  ai  &2  C3  is  -}-,  (Xg  61  Cg 
is  — ,  and  a^  hi  c^  is  +.     And  the  expansion  is 


«i 


&2    C2 

-a2!^'^ 

+  a3 

61  Ci 

&3    C3 

1^3    C3 

^>2    C2 

(^) 


and  the  matrix  of  the  third  order  is  made  to  depend 
upon  matrices  of  the  second  order. 

Similarly,  the  expansion  of  the   matrix   of  the   4th 
order  is 


&2  C2  d^ 

hi  Ci  di 

&1  Ci  di 

61  Ci  di 

hs   C3   ^3 

-a^ 

hs  C3  ds 

+  «3 

ho  C2  d2 

--a4 

&2    C2    C^2 

54    O4    ^4 

&4  C4  d^ 

64    C4    ^4 

63    C3    C?3 

which  makes  it  depend  upon  matrices  of  the  3d  order. 


302 


DETERMIKANTS. 


So  also  a  matrix  of  the  nth.  order  may  be  expanded  to 
depend  upon  matrices  of  the  (n  —  l)th  order ;  and  these 
again  upon  those  of  the  (n  —  2)th  order;  and  so  on. 

259.  Reducing  the  matrices  of  (A),  the  determinant 
of  the  third  order  becomes  — 

(^ih^s  —  ai^gCg  +  ctgftgCi  —  a2&iC3  +  a^hiC2  —  agftgCi- 

Comparing  this  with  the  matrix  here  written,  in  which 
the  first  two  rows  are  repeated  in  order  below 
the  matrix,  we  see  that  the  three  terms  aj  62  ^3, 
ttg  63  Ci,  and  ag  bi  Cg,  read  in  the  direction  of 
the  principal  diagonal,  are  +,  and  the  three, 
Us  62  ^1)  ^1  ^3  ^2?  ^^^  ^2  ^1  ^3?  I'Gad  in  the  direc- 
tion of  the  other  diagonal,  are  — . 

This  is  the  rule  of  Sarins  for  expanding  a 
matrix  of  the  third  order ;  in  practice  the  portion  with- 
out the  matrix  is  not  written,  the  operation  being  carried 
on  mentally. 
2 

Ex.1.         121  =2.2.6  +  4.1.3  +  7.1-5-3.2.7 
-2. 5. 1-1. 4.6  =  - 5. 


ai 

&i 

Cl 

a2 

b. 

C2 

% 

h 

C3 

a, 

h 

Cl 

Ct2 

h 

C2 

2  4  71 

121 

3  5  6 

1  4  7 

2  5  8 

3  6  9 

a  1  a 

a  a  1 

\  a  a 

Ex.2.         2  58  =1.5.9  +  2.6.7  +  3.4.8-3.5.7 
-6.8.1-9.2.4  =  0. 

Ex.  3.         a  a\  =  a^  ■\- a^ +  \  -  d:^  -  a!^  -  a^  =  2a^ -^a^ -Ji-l. 

EXERCISE  XIX.  a. 
1.  Find  the  value  of  the  following  matrices  of  the  third  order  — 


1  2  3 

7 

1 

6 

a  h  g 

x\\ 

2  3  1 

ii. 

1 

-2 

4 

iii. 

hh  f 

iv. 

0  X  1 

3  1  2 

3 

-5 

-1 

Q  f  ^ 

0  0  a: 

DETERMINANTS. 


303 


1  X   y 

a  b  c 

1    «2    y2 

vi. 

b  c  a 

vii. 

1    X3   2/3 

cab 

1     1 

1  -1 

-1     1 


a  b  c 
bed 
ode 


2.  Expand 


12  3  4 

13  2  4 
12  4  3 

14  3  2 


8.   Show  that 


X 

X 

1 

1 

X 

1 

X 

1 

X 

1 

1 

X 

X 

1 

1 

1 

(Expand  to  depend  upon  matrices  of 
the  third  order,  and  then  expand 
these. ) 


=  x(l  -  x)8. 


260.  We  see  from  the  preceding  article  that  as  soon 
as  a  matrix  is  reduced  to  depend  upon  matrices  of  the 
third  order,  we  can  write  out  its  expansion. 

We  turn  our  attention  now  to  the  investigation  of 
those  properties  of  the  matrix  which  enable  us  to  expand 
or  reduce  it  more  readily. 


OPERATIONS    ON    THE    MATRIX. 

Let  D  denote  a  determinant,  of  any  order,  with  sym- 
bolic elements,  and  let  A^  be  the  coefficient  of  ai.  Then 
(Art.  2bd>)  Aiy  which  is  called  sl  first  minor  of  D,  is  a 
determinant  of  the  next  order  lower  than  D,  and  contains 
no  a  and  no  letter  with  suffix  1.  Similarly,  let  A2  be 
the  coefficient  of  a2,  A^  of  a^,  etc. 

Then  (Art.  258) 

D  =  a^Ai  -  a^A^  +  a^Ao^  -  +  •••  (-)«-^a„^„. 

261.    To  multiply  the  matrix  by  any  quantity,  m. 

mD  =  maiA^  —  -wiagztg  +  ma^A^  — f-  •  •  • 
But  this  is  the  expansion  of  the  matrix  in  which  mai  is 
written  for  a^  mag  for  ag,  etc.,  throughout  the  a-column. 


304 


DETERMINANTS. 


Therefore,  to  multiply  a  matrix  by  m  we  multiply  every 
element  of  a  column  or  of  a  roiv  by  m. 

Also,  as  multiplying  by  a  fraction  with  unit  numerator 
is  equivalent  to  dividing  by  the  denominator, 

Therefore,  to  divide  a  matrix  by  m  we  divide  every 
element  of  a  column  or  of  a  row  by  m. 


=  etc. 


«i  ^1 

Cj  1 

mai 

h,  c. 

mai 

mbi  mc^ 

Ex.  1.   ' 

m 

^2    ^2    ^2 

a,  h,  c. 

- 

ma.2  ^2  ^2 
ma.^  63  C3 

= 

C?2       ^2       ^2 

«3         ^3         C3 

Jl  1 

13  4 

Ex.2. 

1  i  1 

=  t\ 

2  14  =111. 

1  1  i 

2  3  1 

8  4  2 

2  2  2 

1  1  1 

Ex.3. 

12  4  3 

=  8 

3  2  3 

=  16 

3  2  3 

=  -64. 

4  6  5 

] 

3  6 

1 

2  5 

262.   Let  each  element  of  a  column  be  the  algebraic 
sum  of  two  quantities. 

Thus,  let     ai=pi-\-  q^,  ag  =  P2  +  Q2)  ^^c. 

Then  D  =  (Pi+gO^i- (P2+q2)A+(P3+q3)A-  +  '" 

=  PiA  -P2A  +P3^3  -  +  ••• 

+  \q^A^  -  q^A^  +  q^A^  -+•••} 

=  the   matrix   with  p  written  for  a,    +  the 
matrix  with  q  written  for  a. 

And  the  matrix  thus  becomes  the  sum  of  two  matrices 
of  the  same  order. 


2  3  1 

1  +  13  1 

1  3  1 

1  3  1 

1  3  1 

Ex.1. 

5  2  1 
3  4  1 

- 

4  +  12  1 
2  +  14  1 

- 

4  2  1 

2  4  1 

+ 

1  2  1 
1  4  1 

" 

4  2  1 
2  4  1 

=  4, 

since  the  s 

econd  1 

na 

trix,  having 

tv 

*^o  colur 

nn 

3  ahke, 

vai 

lishes. 

DETERMINANTS. 


305 


2  3  1 

0  +  231 

0  3  1 

5  2  1 

= 

3  +  2  2  1 

= 

3  2  1 

3  4  1 

1  +  2  4  1 

1  4  1 

Ex.2. 


as  the  second  matrix  will  vanish  after  dividing  by  2. 

263.  Let  «!  be  changed  to  ai  +  nbi,  ag  to  ^2  +  7ib2,  %  to 
%  +  ^^^3?  etc.,  throughout  the  a-column ;  and  let  the  value 
of  the  new  matrix  be  noted  by  D'. 

Then 

D'=  (ai  +  nbi)Ai  —  (as  +  ^^2)^2  +  («3  +  ^^3)^3  —  +  ••• 

=  aiJ.i  —  a2^2  +  ^3^3  —  +  ••• 

+  ^{Mi-M2  +  M3-+---^ 
=  D  +  71  times  the  matrix  with  b  put  for  a. 

But  the  matrix  with  b  written  for  a  has  two  6-columns, 
and  therefore  vanishes. 

Hence  D'  =  D,  and  we  have  the  following  important 

Theorem.  The  value  of  a  matrix  is  not  changed  by 
increasing  or  diminishing  any  column  by  a  midtiple  of  any 
other  column,  or  any  row  by  a  multiple  of  any  other  row. 

In  the  examples  which  follow  we  shall  denote  the 
columns  from  left  to  right  by  (7i,  C2,  C^,  etc.,  and  the 
rows  from  above  downwards  by  E^  E2,  B^,  etc.  Then 
R2  +  Ri  indicates  that  we  are  to  add  the  second  row  to 
the  iirst  row,  and  C2  —  nC^  denotes  that  we  are  to  subtract 
n  times  the  first  column  from  the  second  column :  etc. 


Ex.  1. 


15  3  6 

3  14  1 

4  6  17 

— 

2  2  5  2 

1     5 

3     6 

7 

5  17 

0  14 

5  17 

=r2 

1 

-9    3 

0    2 

-9    3 

4 

1  10 

0    8 

1  10 

=  -24. 


306 


DETERMINAKTS. 


1 

a 

h 

c 

1 

a2 

62 

C2 

1 

a^ 

63  c3| 

1 

a* 

6* 

cH 

Here  we  write  S  R^  —  R^  for  i?.^ ;  R.^  —  2R^  for  7?3 ;  and  2  i^j— i?^ 
for  jR^.  The  C^  of  the  new  matrix  being  all  ciphers  except  1,  the 
matrix  is  at  once  reduced  to  one  of  the  third  order. 

1.  Divide  by  abc. 

2.  i?i,  i?2  —  Rv  ^3  ~  ■^2»  -^4  ~  -^3- 
Ex.  2.     C  ",  r,  ",  .       3.  Divide  by  (1  -  a)(l  -  6)(1  -  c). 

4.  Cj,  Ci  —  Cg,   0.2  —  Cg. 

5.  Divide  by  (a— 6)  (5  — c),  and  reduce. 

Result,    -  abc(l  -  a)  (1  -  6)  (1  -  c)  (a  -  6)  (6  -c)(c-a). 
Ex.  2  may  also  be  reduced  as  follows  — 

1.  If  a,  &,  or  c  =  0,  the  matrix  vanishes.  .*.  a,  6,  c  are  mono- 
mial factors. 

2.  If  a,  or  6,  or  c  =  1,  two  columns  are  alike,  and  the  matrix 
vanishes.     .-.  a  —  1,  6  —  1,  c  —  1  are  binomial  factors. 

3.  If  a  =  6,  or  6  =  c,  or  c  =  a,  two  columns  are  alike,  and  the 
matrix  vanishes.     .-.  a  —  b,  b  —  c,  c  —  a  are  binomial  factors. 

Since  the  expansion  cannot  have  any  term  higher  than  of  9 
dimensions,  these  are  all  the  literal  factors.  And  the  expansion  is 
readily  seen  to  be  abc  (a  —  1)  (6  —  1)  (c  —  1)  (a  —  b)  (b  —c)(c  —  a). 


EXERCISE  XIX.  b. 


1.  Evaluate  the  following  determinants  : 


3  13  1 

12  12 
0  10  1 

13  13 


12  3  4 
4  12  3 
3  4  12 
2  3  4  1 


1111 

0  111 

0  0  11 

10  0  1 

2.  lta,\-\-b,\-\  +  c,\...\-d,\... 
a  4th-order  determinant,  fill  in  the  brackets. 


be  the  expansion  of 


3.  If  a  matrix  be  rotated  through  ninety  degrees,  the  rows 
become  columns,  and  the  columns  rows,  but  they  are  not  disposed 
in  the  original  order.     How  does  this  affect  the  sign  ? 


DETEKMINANTS.  807 

4.  If  the  rotated  matrix  of  3  be  turned  over  in  the  plane,  so  that 
the  4th  column  may  become  the  1st,  the  3d,  the  2d,  etc. ,  how  does 
this  affect  the  sign  ?  How  does  the  matrix  now  compare  with  the 
original  ? 

5.  Show  that  to  keep  the  suffixes  in  order  and  permute  the  let- 
ters is  equivalent  to  keeping  the  letters  in  order  and  permuting  the 
suffixes. 

6.  If  Pi  and  p^  denote  the  diagonals  of  a  matrix  of  the  nth.  order, 
show  that  the  sum  of  the  inversions  in  p^^  and  p.^is  ^n{n  —  1). 

7.  A  matrix  of  the  nth  order  has  both  diagonals  of  the  same 
sign  when  n  or  n  —  1  is  a  multiple  of  4  ;  and  of  opposite  signs  in 
other  cases. 


DETERMINANTS    IN    THEIR    RELATION    TO 
EQUATIONS. 

264.    Take  the  set  of  linear  equations,  with  symbolic 
coefficients  — 

a^  +  b^y +  0^=92, 

Let  Ai  be  the  first  minor  of   |  a^  62  ^3 1   with  respect 
to  ai,  A2  with  respect  to  a^j  etc. 

Multiply  the  1st  equation  by  Ai,  the  2d  by  A2,  the  3d 
by  A^,  and  add ;  then  — 

(ai^i  —  a^A^  +  CLzA)^  +  (Ml  —  M2  +  hA)y 

+  (ci^i  -  C2A2  +  03^3)2;  =  giA^  —  g^A^  +  ^3^3- 

But  the  coefficient  of  a?  is  |  ai  &2  ^3 1 . 

The  coefficient  of  ?/  is  |  %  62  ^3 1  with  b  put  for  a,  and 
it  therefore  vanishes. 


308 


DETERMINANTS. 


The   coefficient  of  2;  is    |  ai  62  Cg  |    with  c  for  a,  and 
it  vanishes. 

The  independent  term  is   |  ^1  62  ^3 1 ,   i.e.   \  ai  &2  C3 1  with 
g  put  for  a. 

I  91  h  C3 1 


.'.  X  -. 


I  c^i  h  C3 


Similarly,       y  =  !-^l|^,  and  .  =  j  ^^  ^^  ^^ 


^1  ^2  ^3 


ai  &2  ^3 


Thus,  in  the  solution  of  the  set  of  3  linear  equations 
in  3  variables,  each  variable  appears  as  a  fraction  whose 
parts  are  matrices.  The  denominator  is  common  to  all, 
and  is  the  matrix  formed  by  taking  the  coefficients  of 
the  variables  in  order  as  elements  of  the  matrix. 

The  numerators  are  the  denominators  in  which  one 
column  is  replaced  by  the  independent  terms  of  the 
equations,  the  coefficients  of  x  being  replaced  in  finding 
the  value  of  x,  those  of  y  in  finding  y,  and  those  of  z  in 
finding  z. 

Ex.  To  solve  the  set   2x  +  ^y  +  4:Z  =  16, 

x  +  ^y  +  2z  =  lS, 


Sx  + 


Here 


16  3  4 

13  4  2 

7  11 

2  3  4 

14  2 

3  11 

y  + 

2  16  4 
1  13  2 

3  7  1 


z=    1. 


3  4 

4  2 
1  1 


2  3 

16  1 

1  4  13| 

3  1 

7 

2  3 

4  ' 

1  4 

2 

3  1 

1 

whence 


x  =  l,  y  =  2,  z  =  2. 


265.  Prom  the  nature  of  the  investigation  of  Art.  264, 
it  is  evident  that  the  same  method  of  solution  applies  to 
the  case  of  n  linear  equations  in  n  variables. 


DETERMINANTS. 


309 


So  that  in  the  case  of  any  number  of  linear  equations 
in  the  same  number  of  variables,  we  can,  at  sight,  write 
down  the  values  of  the  variables  in  the  forms  of  frac- 
tions whose  parts  are  matrices.  The  rest  of  the  solution 
consists  in  the  evaluation  of  the  matrices. 

266.  Take  the  set  of  four  homogeneous  linear  equa- 
tions in  four  variables  — 

QiX  -\-  hiU  +  CiZ  +  diu  =  0 
a2X  +  622/  +  ^2^  +  ^2^  =  0 
a^x  +  %  +  CqZ  -\-dsU  =  0 
a^x  +  b^y  +  C42;  +  d^u  =  0  . 
Divide  the  last  three  equations  throughout  by  w,  and 
solve  for  the  ratios  -,    ^,  and 


(^) 


Then,    -  =  -.      - 

u         I  a2  63  a 


u    u 

hc^d^\     y 


u 


■y 


u 

!  0^2  ^3  ^4  I 

I  a^  bs  C4 1 
z 


z 
u 


I  <^2  h  ^4  I 


I  ag  &3  C4 
-  u 


(B) 


"     I  62  C3  ^4  I  I  a2  C3  ^4  I  I  a2  &3  ^4  I  1  a2  63  C4  ^ 

which  gives  the  ratios  x:  y  :  z  :  u. 

Ex.   Given   Sx-\-2y  —  z  =  4:X  —  6y  +  z  =  0,  to  find  the  ratios 
x:y:z. 


X 

y          _        ^          ora^_2/_2r 

2-1 

3-1          3       2  '  ^'  4      7      26 

-6       1 

4       1          4-6 

.. 

.  x:y:z  =  4::7:26. 

267.    Since  the  denominators,  in  (B)  of  Art.  266,  are 
proportional  to  the  numerators,  write  these  denominators 
for  X,  y,  z,  and  u  in  the  first  equation  of  {A)  of  the  same 
article,  then  — 
tti  I  &2  C3  (^4 1  —  &i  I  (^2  C3  ^4 1  4-  Ci  I  ag  ?>3  ^4 1  —  ^1 1  ^2  &3  C4 1  =  0. 


810 


DETERMINANTS. 


Biit  this  expression  is  the  expansion  of  the  matrix 

I  di  h  C3  di  |. 

Hence  the  Eliminant  (Art.  138)  of  a  set  of  homogene- 
ous linear  equations,  having  the  same  number  of  equa- 
tions as  of  variables,  is  the  determinant  formed  from  the 
coefficients  taken  in  order. 

The  vanishing  of  the  eliminant  indicates  that  the  set 
of  equations  is  consistent,  or  that  the  equations  are 
compatible. 

Ex.  1.   Given  the  equations 

2x  +  32/-l=:3x-2?/  +  4  =  ccH-62/-m  =  0, 
to  find  the  value  of  m  that  the  equations  may  be  compatible. 
Writing  a  unit- variable,  z^  in  the  independent  term,  we  have 


'  Sx  —  2y  +  4:z  =  0          whence 
.    x  +  Qy  —  mz^O 

2  3 

3  ^2 
1       6 

-1 

4 

—  m 

=  0, 

from  which  m  =  4/^. 

Ex.  2.   Given    al  +  bm :  hm  +  en :  en -^  al  =  x 
ratios  a:b:  c  and  l-.m-.n. 

V'^y 

to  find  the 

We  have        «^  +  ^^  =  ^"^  +  ^^=  en -\- al  ^  ^^ 
X                y               z 

say. 

Then,                          al  -\-  hm          —x 

1^  =  0 

bm  -h  en  —  yu  =  0 
al  +  e7i  —  zu  =  0 

And  treating  a,  6,  c,  and  u  as  variables, 
a  —  b  c 


(Art.  266) 


Whence      a:b:  e=mn(x-y-{-z)  :  nl^y-z+x)  :  lm(z-x+y'). 
Similarly,    l:m:  n  =  bc(x-y-hz)  :  ca(y-z-^x')  :  ab(z-x+y). 


m  0  X 

I  0  X 

I  m  X 

m  n  y 

0  n  y 

0  m  y 

0  n  z 

I  n  z 

I    0  z 

DETERMINANTS. 


311 


EXERCISE  XIX.  c. 

1.  Express  the  condition  that  y  =  niiX  +  h^,  y  =  rrioX  +  h^,  and 
y  =  m.^x  +  h^  may  be  satisfied  for  the  same  values  of  x  and  y. 

How  would  you  interpret  this  fact  with  respect  to  the  graphs  of 
the  given  functions  ? 

2.  Express  the  condition  that  Ax-^  +  By^^  4-  C  =  Ax^  +  By^  +  C 
=  Ax.^  +  By^  +  0  =  0  may  be  true,  A,  B^  and  C  being  considered 
as  variables. 

3.  If  3  sheep,  1  cow,  and  4  horses  are  worth  ^  318,  4  sheep,  3 
cows,  and  1  horse  are  worth  -$137,  and  6  sheep,  2  cows,  and  5 
horses  are  worth  $  420,  what  is  the  value  of  2  sheep,  5  cows,  and 
3  horses  ? 

4.  If  a  -{-\(hm  +  gn)=  b  -^  -  (hi  -{-  fn)=  c  +  -  (gl  +  fm)=\, 

I  m  n 

eliminate  Z,  m,  and  w,  and  find  a  cubic  equation  for  determining  X. 

5.  Find  the  condition  that  x^  -{•  ax  =  b  and  x^  +  a^x  =  b^  may 
have  a  common  root. 

6.  By  means  oi  Ax -{-  By  +  Cz  =  0  and  Px+Qy-\-Bz=0  elimi- 

inate  x,  y,  and  z  from  the  statement  -  =  ^  =  -. 

I      m     n 

7.  Determine  m,  n,  and  p  from  the  set 

Wq  —  4m  +  6Wi  —  4n  +  2^2  =  wi  —  4wi  +  6n  —  4w2+i) 

=  Wj  —  4  n  +  6 1/2  —  4  J9  +  ^3  =  0. 
(If  Wq,  i^i,  u^^  ^«3,  be  four  terms  of  a  difference  series,  w,  n,  and 
p  are  intermediate  equidistant  terms.) 

8.  Find  the  eliminant  of  a%-2  +  b'^y-'^'  +  c^z-^  =  alx-^  +  bmy-^ 
+  cn^~2  =  a?'x~2  +  bm'y^  +  cri'^"^  =  0. 


9.  Show  that 


a  -h  b  —  c 
a 
b 


10.   Solve  the  equation  in  x, 


c               c 

a  b  c 

-  c  —  a         a 

= 

b  c  a 

b         c  +  a  —  b 

cab 

«!  +  b^x  Ci  di 

a^  +  b^x  C2  (?2 

=  0. 

«3  +  h^  <^3  ^3 

312  MISCELLANEOUS   EXERCISES. 


MISCELLANEOUS  EXERCISES. 

1.  If  X  +  y  =  a,   y-\-  z=:h,   z-\-x=c,   and  a"^  -\- h'^  ■{■  d^  =  0, 
then  xy  +  yz  -\-  zx  =  I  {ah  +  &c  +  ca). 

2.  Extract  the  fifth  root  of  78502725751. 

3.  Expand  ^-^^ — |-  ^-i_  into  an  ascending  series. 

x-\-l     X  —  1 

5.  Given    x  -\-  by  —  az  =  -^    aic  +  2/  —  -  =  «^,    -  -{-  ay  —  z  =  1, 
to  solve  the  set.  ^  a  a 

6.  Find  all  the  positive  integral  solutions  of  17  x  +  31  ?/  =  100. 

7.  Expand  —^ in  ascending  powers  of  x. 

1  —  x-{-  x^ 

8.  Divide  100  into  two  parts  such  that  twice  the  square  of  one 
added  to  three  times  the  square  of  the  other  may  be  a  minimum. 

x^  —  or  4-  1 

9.  Determine  if  ^^— -    has  a  maximum  or  a  mmmium 

X2  +  X  +  1 

value,  or  both  for  real  values  of  x. 

10.  Factor  3  x^  —  10  x  —  25  and  thence  show  for  what  values  of 
x  the  expression  is  positive. 

11.  Show  that  f^  +  ^  +  ^U^  +  5  +  ^W  9. 

\a      b      cj  \x     y     z] 

12.  Find  the  value  of  Vx^  +  ax  —  x  when  x  =  cc. 

*  ( CL  -4-  x^(b  -4-  x^ 

13.  Find  the  maximum  or  minimum  value  of  ^^ — — — ^-^ — — — ^— 

X2 

14.  Write  the  nth  term  of  1  -  2  x  +  5  x2  -  8  x^  +  H  x*  ... 

15.  Solve  the  set,  x^  +  ^/^  +  x  +  2/  =  14,  2(x2  +  t/^)  +  3x?/  =  29. 

16.  A  person  distributes  50  cents  among  beggars,  giving  7  cents 
to  some,  and  12  cents  to  others.     How  many  were  there  ? 


MISCELLANEOUS    EXERCISES.  313 

17.  A  and  B  play  four  games,  each  having  the  same  sum  at 
starting.  In  the  first  game  A  wins  ^  of  B's  money.  In  the  second 
B  wins  .f  20.  In  the  third  A  wins  J  of  what  B  has ;  and  in  the 
fourth  B  wins  1 5,  and  has  then  J  as  much  as  A.  What  did  each 
start  with  ? 

18.  Find  three  numbers  in  A.  P.  whose  sum  is  21,  while  the  first 
and  second  terms  are  together  equal  to  f  of  the  second  and  third. 

19.  Sum  to  n  terms  the  series  whose  wth  term  is  n'^  —  n  +  \. 

20.  A  falling  body  descends  \f  feet  the  first  second,  f /  feet  the 
second  second,  |/  feet  the  third,  and  so  on.  How  far  will  it  fall 
in  the  wth  second  ?     How  far  in  n  Seconals  ? 

21.  Solve  2x3  —  5c2  —  2ic+l=0  by  putting  it  under  the  form 

x^  —  (x^  -\-  px  -\-  qY  =  ^. 

22.  If  2  w  —  10  is  the  nth  term,  how  many  terms  will  make 
1*2  _  3  6  ? 

23.  Find  three  numbers  in  G.  P.  whose  sum  is  13,  and  the  sum 
of  whose  squares  is  91. 

24.  The  sums  of  n  terms  of  two  A.  P.  s  are  n(n-\-  1)  and 
-  (  -  —  1  ) .     Determine  if  they  have  a  common  term. 

2^2     y 

For  what  same  numbered  term  is  the  term  in  one  series  8  times 
that  in  the  other  ? 

25.  A  sum  of  $  1000  is  compounded  annually  for  4  years  and 
amounts  to  $  1464. 10.     What  is  the  rate  ? 

26.  Divide  1  into  5  parts  in  A.  P.  so  that  the  sum  of  the  squares 
of  the  parts  may  be  ^^. 

27.  Factor  2  ^2  -  21  xy  -  11  ?/2  -  x  +  34  ?/  -  3. 

28.  If  a'^-\-h'^'.a^-c  =  a^-h'^'.a-  c,  then  a  :  6  =  6  :  c  ;  and 
a-2h^-c={a-  hY/a  =  (6  -  c)2/c. 

29.  If  zcc(x-\-  y)^  and  y  oc ^2,  and  if  when  x  =  J,  2/  =  J»  ^  =  h 
express  z  in  terms  of  x. 


314  MISCELLANEOUS   EXERCISES. 


31.  At  a  game  of  cards  3  are  dealt  to  each  person,  and  each  can 
hold  425  times  as  many  different  hands  as  there  are  cards  in  the 
pack.     How  many  cards  are  in  the  pack  ? 

32.  If  the  number  of  combinations  of  n  things  10  together  be 
the  same  as  the  number  5  together,  what  is  the  number  2  together? 

33.  The  sums  of  n  terms  of  two  A.  P.  s  are  as  11  —  5  w  to  11  +  3  n. 
Find  the  ratio  of  their  sixth  term. 

34.  How  many  different  guards  of  6  soldiers  can  be  formed  from 
a  company  of  30  ?    And  relatively, 

1.  How  often  will  A  be  on  duty  ? 

ii.   How  often  will  A  and  B  be  on  duty  ? 
iii.  How  often  will  A  be  with  B,  without  C  ? 
iv.   How  often  will  A  be  without  B  or  C  ? 

35.  At  what  time  after  T  o'clock  will  the  hands  make  an  angle 
of  a°  with  one  another  ? 

36.  A,  B,  and  C  start  from  the  same  point  at  the  same  moment 
to  travel  around  an  island  34  miles  in  circumference.  A  goes  13, 
B,  7,  and  C,  4  miles  an  hour.  When  and  where  will  they  first  be 
together  again  ? 

37.  Find  the  expansion  of  ( 1  —  ^  ~     j  when  x  =  0, 

38.  In  a  G.  P.,  s  =  73.5682,  r  =  1.1,  a  =  3,  to  find  the  number  of 
terms. 

39.  Expand  vTl  as  a  periodic  C.  F. 

40.  Find  values  of  x  and  y  so  that  314  x  ~  451  y  =  13. 

42.  Divide        a^  (b  -c)+  b^  (c-d)+  c^  (a  -  6) 
by  a2  (6  -  c)+  b^  (c  -  a)+  c2  (a  -  5). 


MISCELLANEOUS    EXERCISES.  315 

43.  Express  a;^  —  1  in  terms  of  u  when  u  =  x  —  \, 

44.  li  s  =  a  +  \  prove  that  a*  +  i  =  §2  (^2  _  4)+  2. 

a  a* 

45.  Put  4  ic^  -  4  ic3  +  5  x2  +  3  under  the  form  A^  -  B^. 

46.  Factorize  2x'^ +  Qy'^ -%xy  +  xz -Zyz '-2x -[-^y  -  z, 

A 

47.  Expand  in  ascending  powers  of  x. 

3  —  X  (1  —  x) 

48.  If  -^  =  a,    -^—  =  6,   -^—  =  c,  show  that 

y-\-z  z-\-x  x-\-y 

x^         _         y2         _  g2 


a(l  — 6c)      6(1  — ca)      c(l  — a6) 

49.  From  each  of  two  towns  45  miles  apart  a  traveller  sets  out, 
at  9  o'clock,  towards  the  other  town,  and  the  rate  of  one  is  5  miles 
more  than  |  that  of  the  other.    They  meet  at  12 :  45.     Where  ? 

50.  If  ^  =  1  +  ix+  |x2  +  i5c5+  ...  and  z  =  \  -  x  +  Ix"^  -  \x^ 
+  Jxi ...,  show  that  x  =  2  (1  — 2/^)+  |  (1  -  yz)'^  +  V  (1  -  yzf-\-'" 

51.  P  is  any  point  on  the  bisector  of  the  angle  AOB,  and  A,  P,  B 
are  in  line.  Show  that  the  sum  of  the  reciprocals  of  OA  and  OB 
is  constant. 

52.  A  gives  to  B  and  C  as  much  as  each  one  has.  B  then  gives 
to  A  and  C  J  as  much  as  they  have,  and  then  C  gives  to  A  and  B 
1  of  what  they  then  have,  when  A*has  72  cents,  B  104  cents,  and 
C  28  cents.     What  had  they  at  first  ? 

53.  xy  =  c{x-{-  ?/),  yz  =  a(y  -\-  ^),  zx  =  b(z  -{-  x),  to  solve  in  x, 
y^  and  z. 

54.  If  x(^B-^b)=h-^b,  find  the  value  of  x(B  -b)+  Bh 
in  terms  of  P,  6,  and  h. 

55.  If  a  carriage  wheel,  16 1  feet  in  circumference,  took  one  sec- 
ond longer  to  revolve,  the  rate  of  the  carriage  would  be  1|  miles 
less  per  hour.  How  fast  does  the  carriage  go  ?  Explain  the  double 
solution. 


316  MISCELLANEOUS   EXERCISES. 

66.  Prove  by  inequalities  that  the  triangle  of  greatest  area,  with 
given  base  and  perimeter,  is  isosceles. 

57.  If  xy  +  yz  +  zx  =  0,  and  xyz  =  a,  find  y  and  z  in  terms  of 
a  and  x. 

58.  Resolve  a;^  +  6a;-2 .^^^  partial  fractions. 

(x2-x  +  l)(x-l)2 


59.   By  the  expansion  of  (1  +  1  —  x)»»  show  that 
Ci(l  +  Ci  4-  C2  +  ...)=  2ci  +  4c2  +  6C3  +  . 


60.  By  the  expansion  of  (1  +  x  —  1)^  show  that 

i.    C0-C1  +  C2-  Cb  ..•  Cn  =  0. 
ii.   ci  ~  2  C2  +  3  C3  —  4  C4  •••  =  0. 

61.  If  Pand  Q  are  the  pth.  and  gth  terms  of  an  A.  P.,  what  is 
the  (p  +  g)th  term  ? 

62.  Divide  a  number  into  two  parts  such  that  the  product  of 
the  parts  shall  be  equal  to  the  difference  of  the  squares  of  the 
parts. 

63.  Three  numbers  are  in  A.  P.  If  1  be  taken  from  each  of  the 
first  two,  the  three  terms  will  be  in  G.  P. ;  and  if  the  last  term  be 
increased  by  the  first,  they  will  be  in  H.  P.     Find  the  numbers. 

64.  Eind  the  square  root  of  2x  +  2^{y(2  -  y)+ x'^  -  1}- 

65.  How  many  sums  can  be  made  of  a  farthing,  a  penny,  a  six- 
pence, a  shilling,  a  half-crown,  a  crown,  a  half-sovereign,  and  a 
sovereign  ? 

66.  A  stream  runs  2  miles  an  hour,  and  a  rower  who  rows  4 
miles  an  hour  wishes  to  go  directly  across  it.  At  what  angle  up 
the  stream  must  he  direct  his  boat  ? 

67.  Find  the  approximate  value  of 

{ Vr+^  +  v'(l  +  iC)2}  ^  {1  +  X  +  VTTx] 

when  first  powers  of  x  only  are  retained. 


MISCELLANEOUS   EXERCISES.  317 

2 S  or  -A-  x^ 

68.  Expand — to  the  term  containing  x^. 

69.  Find  the  fraction  which  expanded  gives 

2  -  Sx  -\-  x'^  +  2 x^  -  S  x^  +  x^  +  2 x^ 

70.  Show  that  1  +  ^  +  ^^  +  ^^+-  ^  1±^. 

1—X  +  X^  —  X^+"'        1—x 

71.  Express  x^  —  5  x^^y  4-  9  x^y^  —  7  x^y^  +  2  ic?/*  —  2/^  as  a  func- 
tion of  ?/  and  ^,  when  z  =  x  —  y. 

72.  A  rectangle  is  inscribed  in  a  triangle,  and  has  a  side  coinci- 
dent with  the  base  of  the  triangle.  Show  that  the  area  of  the 
rectangle  is  a  maximum  when  its  altitude  is  one-half  that  of  the 
triangle. 

73.  A  cubic  foot  of  lead  weighs  715  pounds.  If  10  pounds  of 
lead  is  formed  into  a  cubic  block,  how  long  is  the  edge  ? 

74.  What  three  linear  expressions  divided  into  x^  —  7  x  -1-  10 
will  each  give  a  remainder  4  ? 

75.  Express  x^  +  S  x'^y  —  6  xy^  —  2  ?/3  as  a  function  of  y  and  ^, 
when  z  =  x-\-  y. 

76.  If  ic2  -  2  a;  -  1  =  3,  find  the  values  of  x^-  5  xH  3  x'^-  x+l 
as  a  linear  function  of  x. 

77.  If  Bac  —  5A,  Cab  =  cA,  find  the  value  of 

a^BC 

2(B^/l  -  (72  +  CVl  -  ^2) 
in  terms  of  abc  and  A. 

78.  Put  x^  -\-  x^(a  +  h)—  x{a^  —  ah)  —  a'^b  into  linear  factors. 

79.  If  £+^  +  «^«  =  ^lL^+6^^,  provethat  a  =  6. 

a  —  X     a  -{-  X     b  —  X     b  -\-  x 

80.  If ^ = ^ = ^ ,  then 

lx{ny  —  mz)      my(lz  —  nx)      nz(mx  —  ly) 

^(Z-x)+— (m-2/)  +  — (n-c)  =  0. 
Ix  my  nz 


318  MISCELLANEOUS  EXERCISES. 

81.  Two  circles,  whose  diameters  are  D  and  df,  are  made  to  over- 
lap until  their  common  chord  is  c.  Find  the  distance  between  the 
centres,  and  explain  the  quadruple  solution. 

82.  Two  perpendiculars  from  a  point  to  adjacent  sides  of  a 
square  are  6  and  7,  and  the  line  to  the  vertex  not  related  to  those 
sides  is  6.  Find  the  side  of  the  square,  and  explain  the  double 
solution. 

83.  The  wheel  of  a  barrow  is  20  in.  in  diameter,  and  its  axle  is 
1  in.  out  of  the  centre.  If  the  wheel  turns  uniformly  at  the  rate 
of  1  revolution  per  second,  find  the  greatest  and  least  velocity  of 
the  barrow  in  feet  per  second. 

84.  A  V-shaped  trough  has  an  angle  of  60°,  and  is  6  feet  long, 
A  sphere  12  inches  in  diameter  is  placed  in  it  and  rolls  throughout 
its  length.     How  many  revolutions  does  it  make? 

85.  If  a  +  X  Vl  +  a^  =  a  Vl  -  x^  +  x  Vl  -  a^  fmd  x. 

86.  From  a  log  20  in.  diameter  and  20  ft.  long,  the  largest  beam 
is  to  be  cut  so  as  to  be  twice  as  wide  as  thick.  How  many  cubic 
feet  will  it  contain  ? 

87.  Given  x2  -f  i  =  12  +  6  f  x  +  -V  to  find  x. 

x^  \        xj 

88.  Given  ^^(x^  -f  y"^)  =c^(p  —  Ix  —  myy,  to  find  n  in  terms  of  c, 
when  the  terms  of  two  dimensions  in  x  and  y  form  a  square. 

89.  To  go  through  a  rectangular  field  along  a  diagonal  is  10  rods 
shorter  than  going  around,  and  one  side  of  the  field  is  twice  the 
other.     Find  its  area. 

90.  Divide  a  into  two -parts,  such  that  the  difference  of  their 
squares  divided  by  the  sum  of  their  squares  is  a  max.  or  a  min. 

91.  Given  x(i-^x  +  iy=  102(x  +  V^)  -  2576,  to  find  x. 

92.  If  two  chords  of  a  circle  intersect  at  right  angles,  the  sum 
of  the  squares  on  the  segments  is  equal  to  the  square  on  the 
diameter. 


MISCELLANEOUS   EXERCISES.  319 

93.  Find  the  nth  term  of  the  A.  P.  whose  sum  to  n  terms  is 

94.  A  pair  of  wheels,  whose  diameters  are  d  and  d'^  are  rigidly- 
fixed  upon  an  axle,  a  feet  apart.  How  large  a  circle  will  the 
smaller  wheel  describe  when  the  set  rolls  on  a  plane  ? 

95.  ABCD  is  a  square  with  side  s.  AE  =  BF=  CG=DH=ns, 
E  being  on  AB,  F,  or  BC^  etc.  Determine  n  so  that  the  square 
formed  by  AF,  BG,  CH^  and  DE  may  have  a  given  area  cfi. 

Explain  the  double  solution,  and  thence  find  the  maximum  and 
minimum  values  of  a^. 

96.  A  farmer  bought  some  sheep  for  |72.  If  he  had  bought 
6  more  for  the  same  money,  he  would  have  paid  $  1  less  for  each. 

How  many  did  he  buy  ?  Explain  the  double  solution,  and  change 
the  wording  of  the  question  accordingly. 

97.  A  G.  P.  and  an  A.  P.  each  has  its  first  term  =  a,  and  the 
sums  of  the  first  three  terms  are  equal. 

Find  r  in  terms  of  a  and  d,  and  show  that  both  values  of  r  satisfy 
the  conditions. 

98.  If  a,  &,  c  are  all  positive,  ?i  +  1  +  1  >  ^'^  -f  ^2  _|_  ^2  ^^^ 
^Va+V6+Vc  a     h      c  abc 

Vabc 

99.  If  u  =  l  +  x  +  x^+'"  Siiid  v  =  l-x  +  x'^-+  ..-,  1  +  1  =  2. 

u      V 

100.  Show  that  l-(ll'  +  m7n'  +  nn'y^=(mn'-m'nyi-{nV-Vny 
+  (Im'  -  Vmy\  if  P  +  m2  -\- n^  =  V^  +  m'--^  +  ?z'^  =  1. 

101.  X  and  Y  are  towns  20  miles  apart.  A  person  goes  a  certain 
number  of  miles  in  a  certain  direction,  and  then  changing  his  course 
through  a  right  angle  arrives  at  Y  after  travelling  1  mile  further 
upon  the  second  course  than  the  first.  How  long  was  he  in  going 
from  X  to  Y  at  4  miles  an  hour  ? 

102.  On  the  rectangle  ABCD,  with  sides  AB  =  a  and  BC=  6, 
P,  Q,  i?,  S  are  taken,  so  that  AP  =  na  =  CB,  and  BQ=  nb  =DiS, 
Show  that 


320  MISCELLANEOUS  EXERCISES. 

area  of  Rectangle  :  area  of  Parallelogram  PQBS  =  1:2 n^— 2/1+1; 
and  explain  the  result  when  n  =  1.     When  n  is  negative. 
Has  the  parallelogram  any  maximum  ?  any  minimum  ? 

103.  n  /i  =  l  +  ic  +  -  +  ^  +  ...,  prove  that 

^2  +  1_^ /.     ,    X2    ,    CC*        X6 


/l+^%^  +  ^%  ...I. 

1         2!      4!      6!  / 


104.  A  ladder  20  feet  long  leans  against  an  upright  wall,  and  has 
its  foot  3  feet  from  the  wall.  If  a  person  pulls  the  foot  outward, 
compare  the  rates  with  which  the  foot  and  the  top  begin  to  move. 

105.  A  room  18  by  24  is  to  be  so  carpeted  as  to  have  its  floor 
two-thirds  covered,  and  the  width  of  the  uncovered  part  is  to  be 
uniform  around  the  room.     Find  the  size  of  the  carpet. 

106.  In  a  rectangular  garden,  40  by  60  feet,  a  flag  pole  50  feet 
high  is  placed  at  6  feet  from  a  longer  side  of  the  garden  and  15  feet 
from  an  adjacent  side.  How  far  is  the  top  of  the  pole  from  each 
corner  of  the  garden  ? 

107.  If  «&2  _  5c2  =  a,  hc^  -  ca"-  =  6,  ca2  -  ah^  =  c,  show  that 

(ab  +  6c  +  ca)(a'^  +  52  +  ^2)  =  -  (a*  -f  6*  +  c^). 

108.  Find  all  the  values  of  x  from  /x  +  iV+1^:-- 

V        xj  x^ 

109.  Two  wheels,  in  the  same  plane,  are  3  feet  and  1  foot  in 
diameter,  and  their  centres  are  4  feet  apart. 

Find  the  length  of  the  belt  which  envelops  the  wheels  and 
crosses  between  them. 

110.  Show  that  l+x  +  x2  +  ic3^--fl  +  l  +  -l+  ...V 

x\        x     x^  J 

111.  If  (^^±yiy=  lizl!,  then  each  fraction  =  1. 

\y  +  zxj       1  -  x^ 

112.  Express  as  a  fraction  1  —  2  ic  (1  —  x)  (1  +  ic2  +  x*  +  •••)  + 
I        3      I.2V3J       1.2.3^3/  r 


MISCELLANEOUS   EXERCISES.  321 


113.  Given 
a 


{x  —  d)(x  —  c)      {a  —  c){a  —  x)      {c  —  x)(c  —  a)      a  —  c 
to  find  X;  and  also  the  ratio  (c  +  a)2  :  (ex  +  a"^). 

114.  Determine  the  isosceles  triangle  in  which  the  altitude  is 
equal  to  one-third  the  whole  perimeter. 

115.  The  radius  of  a  circle  being  r,  find  the  angle  between  two 
radii  when  the  triangle  formed  by  them  and  the  chord  through  their 
extremities  is  a  maximum. 

116.  Through  a  point  P  to  draw  a  line  so  as  to  form  with  two 
given  intersecting  lines  the  minimum  triangle,  show  that  there  are 
two  minima,  and  explain. 

117.  A  man  walks  from  A  to  B.  If  he  had  walked  m  miles  an 
hour  faster  he  would  have  been  h  hours  less  on  the  road,  and  if  he 
had  walked  m^  miles  an  hour  slower  he  would  have  been  h^^  hours 
longer. 

Find  the  distance  from  A  to  B,  and  the  rate  of  walking. 

118.  If  X  =  re,  y  =  rs,  r  =  as,  and  s^  +  c^  =  1,  eUminate  c,  s, 
and  r. 

119.  How  many  sets  of  positive  integers  satisfy  the  system 

120.  A  rectangular  garden,  sides  a  and  ?>,  is  to  be  bordered  with 
a  walk  of  uniform  width  which  shall  occupy  one-half  the  plot. 
Find  the  width  of  the  walk ;  and  explain  the  double  solution. 

121.  If  f(x)  ~l+^  +  f  +  f^  ...,  find  /(O  +  /(_  ^),    where 

1       Z\       o! 


122.  If  s,  p,  g,  be  the  sum,  product,  and  quotient  of  two  num- 
bers,  ;?  =  S2(g  _  2g2^  3g3_4g4  _| ), 

123.  Two  wheels,  A  and  B,  geared  together  should  move,  as 
nearly  as  practicable,  with  relative  velocities  of  1401  and  1945. 


322  MISCELLANEOUS   EXERCISES. 

The  number  of  teeth  in  a  wheel  being  limited  to  not  more  than  120, 
find  the  numbers  to  be  employed. 

After  100  revolutions  of  A  how  much  will  B  be  in  advance  of, 
or  behind,  its  true  place  ? 

124.  The  resistance  to  sliding  a  stone  on  the  ground  varies  as 
the  weight,  and  the  weight  varies  as  the  cube  of  the  diameter. 
The  power  of  a  running  stream  to  move  a  stone  varies  conjointly 
as  the  square  of  the  diameter  and  the  square  of  the  velocity  of  the 
stream.  Show  that  if  the  velocity  of  a  stream  be  doubled  it  can 
move  a  stone  2'^  times  as  heavy  as  before. 

125.  If  a,,  a.^,  a.^,  a^  be  perpendiculars  from  the  vertices  of  a 
square  to  any  line,  and  p  be  the  perpendicular  from  the  centre  of 
the  square  to  the  same  line,  show  that  2a^  =  4j)2  _(-  §2^  where  s  is 
the  side  of  the  square. 

126.  If  &p  ^21  &3,  ?>4  be  line-segments  from  any  point  to  the  ver- 
tices of  a  square,  and  p  be  the  line-segment  to  the  centre,  show 
that  26-2  =  4j9'2  +  2  si 

127.  The  natural  numbers  are  grouped  as  follows  : 

(1)  (2,  4)  (3,  5,  7)  (6,  8,  10,  12)  (9,  11,  13,  15,  17)  ... 
Show  that  the  sum  of  the  numbers  in  the  nth  group  is 


128.    If  xy  :=1,  show  that  a;"  +  ?/"  : 


(x+2/)«-4    0.        1 


TABLE   or  PRIME  NUMBERS   UNDER  1000. 

The  hundreds  are  found  in  the  top  row,  and  the  two  remaining  figures  in  the 
body  of  the  table. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

I 

01 

II 

07 

01 

03 

01 

01 

09 

07 

2 

03 

23 

II 

09 

09 

07 

09 

II 

II 

3 

07 

27 

13 

19 

21 

13 

19 

21 

19 

5 

09 

29 

17 

21 

23 

17 

27 

23 

29 

7 

13 

33 

31 

31 

41 

19 

33 

27 

37 

II 

27 

39 

37 

33 

47 

31 

39 

29 

41 

13 

31 

41 

47 

39 

57 

41 

43 

39 

47 

17 

zi 

51 

49 

43 

63 

43 

51 

53 

53 

19 

39 

57 

53 

49 

69 

47 

57 

57 

67 

23 

49 

63 

59 

57 

71 

53 

61 

59 

71 

29 

51 

69 

67 

61 

77 

59 

69 

63 

77 

31 

57 

71 

73 

63 

87 

61 

73 

77 

^Z 

37 

63 

77 

79 

67 

93 

73 

87 

81 

91 

41 

67 

81 

83 

79 

99 

77 

97 

^3 

97 

43 

73 

83 

89 

87 

^^ 

87 

47 

79 

93 

97 

91 

91 

53 

81 

99 

59 

91 

61 

93 

67 

97 

71 

99 

n 

79 

83 

89 

97 

323 


ANSWERS   TO   EXERCISES. 


I. 

a.  2.  i.  -  3.  ii.  -  183.  iii.  30521/415800.  3.  La-  b. 
ii.  a-h  +  b^-  ab'^  +  ab^.  4.  i.  a^ -9b^  +  ^c^  +  4:  ac. 
ii.  2(m2  -  l)a2  4  2(n2  -  1)62.        lo.   4th.        11.  pth. 

b.  4.  i.  12.     ii.  4 J.     5.    1-x.     6.    -5.     7.    800.     8.   2a  +  3  6. 

9.  48,24,36.      10.    $1100.       11.   $120.       12.    120.       13.    67. 

II. 

b.  6.  i.  0.  ii.  UabcZa.  iii.  0.  iv.  abc.  10.  'Za^  +  S'Za'^b 
-^-Qabc.      11.    2a35+2a252_^3  2^25c.       12.    Sa3_f3  Sa^d+O  2a&c. 

C.    1.  a:4_i0x3+35ic2-50ic+24.     2.    a:H2x3-13  x2-14  x+24. 

3.  24a:*  +  154a:H269x2+154x+24.  4.  x3  +  x2Sa  +  x(2  Sa6-2«2) 
+(Sa25  _  2a3  _  2  a^c).  5.  Sa^  -}.  5  Sa^?,  +  lO  Sa^ft^  4.  20  Sa^^c 
+  30  Sa262c.      6.   2a*  +  4  Sa^^  +  6  Sa262  +  12  l^a^bc  +  24  a6cd. 

d.  1.  26,  -46.     2.   x'^-x.     6.   2.71828.     7.    ic*  + a:3-ic2  +  2  x. 

e.  1.14-2x2  +  3x^+14x0  +  25x8.         2.    1.     3.    x2-2x-l. 

4.  mc-^2nab-\-pa^,     5.   -J6(8a+1).     6.  1+x.     7.   1.     8.  x^-l. 

10.  12  +  Ci2  +  C22+--..  11.    C2  +  C1C3  +  C2C4  +  C3C5+---.  12.    m-^m'^ 

4.1^3  _+....  13.  ^  =  i,  J5=r--^,   C  =  ?^^^^-^,  etc. 

14.  3.1416.        15.  3.1416.  a  a  a 

f.  1.  x3  +  2  x2  +  3  X  +  4.  2.  3  a3  +  a2  _  1.  3.  1  -  2  x  +  2  x2. 
4.  l-2a-i+2a-2-2a-3....  5.  Q=x2-2  x+6,  i2=-18x2+18x-7. 
6.  l  +  (3a-a2  +  «3+5^4)(i4.Qj4_|.^8  +  ...).  8.  l-x+x2-x3+ -••.. 
9.  x+2x2+3x3+4x*  +  ...,  and  x-i+2  x-2+3x-3+....       10.  x-1. 

325 


326  AKSWERS. 

11.  l-{-ix-\-ix^  +  ixK  13.  a2(5_c)-a(62_c2)  +  6c(5-c), 
or -(a-&)(6-c)(c-a).  14.  x^-hy\  15.  x(l  +  2x)/ {l-x-hx^). 
16.  6%.  17.  a  +  2  -  2  a-2  -  a-s.  18.  a:*  +  x^  +  3  x2  +  2  a; 

-  2  x-i  -  3  x-2  -  x-3  -  X-*.  19.  4  x3  +  x2  +  X  +  1.  21.  J.  =  a-2, 
^  =  a-3,  0  =  0,  i>  =  -  a-5,  /x  =  -  ic4(a  -  x)/a%a^  -  ax  +  x^). 

III. 

a.  1.  (x-l)(x+l)(x-2)(x+2).  2.  (2a  +  3  5  +  c)(a-5-c). 
3.  (x^  -  px  +  l)(ic2  -qx-1).  4.  (x  +  y  +  a)(x  +  ?/  +  5). 
5.  (a  + x-l)(a  +  x-2).  6.  169  xy.  7.  (a  +  b -\-  c){a  +  b  -  c) 
(&  +  c-a)(c+a-6).        8.   (3x-l)(4x-l)  and  3(x-2)(2  x-3). 

9.  (x-l)(x  +  l)(x'^ -px  +  q). 

b.  1.  i.  (6  H-c)(c  +  a)(a  + 5).  ii.  (&  -  c)(c  -  «)(a  -  5). 
iii.  (5  —  c)(c  —  a)(a  —  6).  iv.  (a  —  b)(b  —  c)(c  —  a)(a -{- b  +  c). 
V.  vanishes.  vi.  (a  —  b)(b  —  c){c  —  a)(1>a'^  +  'Zab), 
vii.  (a  —  6  +  1)(6  —  c  +  l)(c  —  a  +  1).  viii.  —  2  abc.  ix.  van- 
ishes. 2.  (a2  _  be)  (b^  -ca)  (c^  -  ab).  5.  (x  -  1 )  (x  -  3)  (x  +  5) 
(x  +  7).  6.  (x-f-2)(x  +  6)(x2  +  8x  +  10).  7.  (x  +  V3) (:k  -  V^) 
(x+V2)(a:-V2).      8-  24a?)c.      9.   (a  +  5-c)(?)  +  c-a)(c+rt-6). 

10.  (a-&)(6-c)(c-(^)((^-a).  11.  abc.  12.  (a-6)(6-c). 
13.   (x  -  a  +  5)  (x  -  &  +  c)  (x  -  c  +  a) . 

c.  1.  a  +  6  -  1.  2.  (X  +  2  1/  -  2)2._  3.  (b  -  c)\/a. 
5.  x2-2«x  +  a2+52.  7.  i.  {x  -  J(3  +  V21)}{x  -  i(3-V21)}. 
ii.  (x-l  +  20(x-l-20.  iii.  (2x  -  1  +  V3)(2  x  -  1  -  V^)- 
iv.  5(x+x%4-tV^129)(x+A-tVv'129).    v.  a(x+-i+J-Vr=4^2] 

/a;  +  J__ J_Vl--4a2y    vi.  (x-l)(i9x-l).     vii.  (o^-x+l) 

(a  +  6  •  X  +  !)• 

IV. 

a.  1.  i.  x2  -  2  X  -  1.  ii.  x2  -  3.  iii.  2  x2  -  3  x  +  3.  iv.  5  x. 
V.  x2  -  1.  vi.  a26-2  -  1  +  52^-2.  2.  a2  -  62  =  40.  3.  c  =  -  4, 
=  -3J.  4.  (cp-ar)2=:(6;)-«^)(cg-6r).  5.  {BG  -  Aq) 
iCQ  -  BB)  =  {C^  -  ABY,  where  ^  =  a-ai,  B  =  b-bu 
C=c  — Ci,   §  =  cai  — aci,  i?  =  c6i  — 6ci.       6.  a  =  ^,   =  —  2. 


ANSWERS.  327 

b.  1.  i.  12(x-2)(x-3)(a:  +  3).  ii.  (x  -  a)(x  -  b)(x  ~  c). 
iii.  i^p^^pi,  3.  i.  2332.      ii.  22325.      iii.  23103.      iv.  23131. 

V.  2^32527.  4.  223.  5.  2520.  6.  5,  150.  7.  x-2, 

8.  4,  X  -  3. 

V. 

a.  1.  i.  (2x  +  i/)/(2x-y).  ii.  (Sx -{■  y)/(Sx -y). 
iii.  (iix -\- y)  /  (nx  —  y) .  iv.  2(a  —  x)(a2  + x2)/ax2.  v.  a  +  h  +  c, 
vi.  l/(2x- —  1).  vii.  0.  viii.  1.  ix.  a  +  5  +  c.  x.  1.  xi.  0. 
xii.   —1.     xiii.  x.    xiv.  a-\-h-\-c.    xv.  —2.    xvi.  0.    3.  a.     6.  r/f. 

b.  1.   -  f.  2.  00.  3.  00,   (c2  +  2  ac  -  a2)/2  c ;  -  00  ;  -. 

5.   (a2-f2)/(4a  +  5).     6.  00,  2ab/(a  +  b);  00.    7.  00.     8.  0,  V^- 

9.  0,  1,  i(l±Vl7).  10.  a  =  b.  11.  8,  18.  12.  ab/(a-b). 
13.  29.5....  14.  X  =(b'^  -  a^)  /  (2  a  -{-2  b  -I),  15.  00,  1. 
16.  X,  4.        17.  3/2  a2.       is.  36,  12,  16.       19.  $3.75.       21.  32. 


VI. 

a.  1.  0,  4,  16,  00,  -  32.  2.  144  :  125.  3.  (2  ajS  -  a  -  /3) 
/(a  +  ^-2).  4.  15:8.  5.  (n/m)^.  6.  1,  -3,  f. 
8.   (?7i2+n2)  :  (m2-n2),  {2  n2  :  (m2  -  n2)}2.     9.  0,  -1.    10.  -1  :  x2. 

b.  1.  s  =  16  ^2.  2.  145/416.  3.  X  =  ±  4.  4.  x  =  0,  6. 
5.  (x-yy  =  ^\xy.  6.  6,4.  9.  10  hrs.  29.74  min.  10.  1666|. 
11.  ©  :  D  =75  X  (28)2  :  (99)2.     12.  0.027  in.    13.  5 hrs.  31  m.  nearly. 

VII. 

2n-5  _^  PI 

a.  1.  a.  2.  -^z.  3.  X  2  .  4.  a  ^.  6.  x^?.  6.  (ft^/a"*)"*". 
7.  ggy/g  -f  a-2v^2  _  2.  8.  a^  +  b^.  9.  x^  +  rx^  +  3fV  ^'  =  0- 
10.  (aT^^-2a)2  =  4(a2_a;2).  n.  2.  12.  n  =  {.  13.  w  =  -6. 
14.  m"-i  =  n.  15.  n  =  S.  16.  x2  +  2  +  8  x-i  +  27  x"^  ; 
i2  =  97  x-2  +  55  x-3. 

b.  1.  i.  1  - V5.  ii.  -24J.  iii.  ^38  +  13^6).  iv.  \/2a". 
V.  30V2_+_3V2.  vi.  9bVJa.  vii.  8V2.  viii.  a?)(3  a  +  5  6)  \/2a'. 
2.  i.  3V2a/4a.    ii.  l+V^.    iii.  V^-    iv.  ^(3V5-5)VlO+2v5. 


328  ANSWERS. 

V.  1  +  v/2.    vi.  2  -  ^3.    vii.  1(2  -  ^2  -  ^6).         3.   The  former. 

4.  The  latter,  by  JO'^V^  -  ^V^)-  5.  K^^  -  ^v^^  +  2^2). 
6.  fv2(V2-^2)(2+^4  +  ^2). 

C.    1.   V^  +  \/3-       2.  2  +  i.       3.  1  +  V'3.       4.  tf  +  V(l  -  w^)- 

5.  l-^.      6.  a^-?/+V(^-2/''):      7.  2+V3-V2.      8.   Vl^+Vl^. 

9.  l  +  iV2.  10.  1(V16  +  VQ).  11.  (a  +  Va2-4)/V2.  12.  3. 
13.   V(6  +  2i)). 

VIII. 

a.  1.  100(s-h)/b,  2.  mnr/{m-n).  3.  ^  =  2  a  +  3  &, 
^  =  a+3&.  4.  A's  =  J(m  +  w +p),  B's  =  i(m-n +^). 

5.  ah /{a  ~h).  ^.  AO  =  l\\a^  where  a  is  the  distance  be- 
tween consecutive  points.  8.  a'^/d^.  9.  24  and  30. 

10.  ac{hi  —  ai)/(abi  —  ai&),  bc(bi  —  a{) / {ahi  —  aib).  11.  20 
miles  an  hour  and  704  ft.  long.  12.  dist.  =  ha(h  —  b)/h  ; 
rate  =  a(h  —  b)/b. 

c.  1.  I  a2.  2.  i.  dm/  Vm^  +  n'^,  dn/Vm^n^. 
ii.  d2m?i/(m2+n2).  iii.  dmn/(m'^+n^).  iv.  d{m'^—n^)/(m'^-\-)v^), 
3.  i.  ab/Va^  +  b^  ii.   (a2  -  62)  /  Va^  +  6^.           4.    62/V^i2Tn". 

6.  iVl5.       6.    64.  7.   4in.  rad.       8.    ^(ir^-1).       9.    iV^- 

O  Q  

10.   _ths  nearly.        11.   _ths  nearly.          12.  i.   w2.      ii.  |Vl07 
71  n  

13.  V3-  14.  1:2.  15.  i.  V29 :  5.  ii.  \/29 : 2.  16.  t  +  ti'.t-h. 
17.   ladder  =  39J  ft.  nearly  ;  position,  34J  ft.  from  wall. 

d.  5.  3,  1. 

IX. 

a.  1.  i.  —\b,  }c.  ii.  lia  —  b),  J(a  +  5).  iii.  a/b,  b/a. 
iv.  HiV2,  i-iV2.  V.  l/(a-6),  l/(a  +  6).  vi.  i-iV2, 
1+1^2.  2.  i.  b  =  a/(\-\-a).  ii.  6=:a/(l  +  a).  iii.  a6  =  1. 
3.    cx2+6a:+a=0.^_5^J<V#+8^±(^).       6.  BP=^(2  a^-s^). 

7.  J5P=i(s±V5s2_8a2).         8.  ^0  =^{4^^  ±^^*  -  16  a*)}- 

9.  Z2  =  5«2.        10.    00=iZ(V5-l). 

b.  9.  i.  min.  —  |.     ii.  min.  —  |.     iii.  max.  4J.     iv.  min.  —  2J. 

10.  |.  12.  1.  13.  a/(l  +  w),   n«/(l  +  n),   wa2/(i  +  n). 

14.  J  a,  J  a.  16.  imag.  22.  min.  J  s'^.  23.  max.  |ths  of 
the  square.    24.  J  Z2.     25.4^5,8^5.     26.5^2-     27.4^5,8^5. 


ANSWERS.  329 

28.  10  -  2  v3  from  A.  29.  p'^/a'^  +  q^/b'^  =  1.  30.    IJ,  3^. 

31.   i.   —p.     ii.  q.      iii.  p^  —  2  q.      iv.   —p/q.      v.   {p^  —  2q)/q'^ 
vi.  iipq—p^.        32.    11  days. 

C.  1.  i.   (b-ay/2b,  00.  ii.  oo,  b(b  -  2  a)/2(b  -  a). 

iii.  00,  (b-ay^/(2a-b).    iv.  0,  fa.     v.  oo,  (5  a6  + 6'^  -  a-^)/2a. 

2.  0,  V{(2«-&)/«2&}.     3.  0,  JV3.    4.  faV^-    5.  ia(5^-6"^)2. 
6.  0,  4.  7.   16,  25.  8.  0,  -  2  a(  Vl  +  a2_  Vl  -  a^) 

/{a2  +  2  -  2\/l  -  a*}.     9.   f ,  -  f    10.  0,  «,  K«  ±  VSa^-Baft). 


a,  1.  1.  a;  =  29  +  7p,  ?/  =  2-3jt).  ii.  a;  =  l  +  17p,  y=ll-lSp. 
iii.  x  =  13p  +  2,  ?/  =  45^  +  4.  iv.  ic  =  48  +  llp,  2/=i).  2.  2  and 
3,  4  and  6,  6  and  9.  3.  5  7-in.  and  1  13-in.  4.  6  4-lbs.  and  3 
7-lbs.  5.  63  ;  general  formula  420^9  -  357.  6.  i.  21  wide,  33 
narrow,    ii.  26  wide,  25  narrow,    iii.  41  wide,  1  narrow. 

h,  I.  x  =  6,  y  =  2.  2.  x  =  S,  y  =  6.  3.  x  =  a  —  abc^/(b -\- ac), 
y  =  b-i/{b  +  ac).  4.   X  =  aV(a  -  6),  y=  b^/(b  -  a), 

5,  2sd=(z  +  a)  (z-a  +  d).         6.  A  =  200,  B  =  300.        7.  xV 

8.  93.  9.  alb  -  c) / (b  -  a)  of  the  first,  b{a  -  c) / (a  -  b)  of 
the  second.         10.  150  acres,  rent  =  ^  600. 

C.  1.  X  =  7,  2/  =  5,  ^  =  3.  2.  X  =  2,  2/  =  0  =  1.  3.  a;  =  3, 
y  =  2,  ^=— 1.  ^.x  =  y  =  b,z  =  0.  5.   x=(b  —  a) 

/{a  —  b){b  —  c)(c  —  «),  with  symmetrical  expressions  for  y  and  z. 

6.  ic  =  J,  2/  =  i,  ^  =  i-  7.  3  a  =  6  +  5  c.  8.  5  6  =  3  c. 

9.  7  a  +  6  +  11  c  =  0.  12.  x:y:z:u  =  1:S  :  I:  S.  IZ.  x  =  z  =  ^^, 
y  =  1.  14.  a  =  1,  5  =  3,  c  =r  2,  d  =  4.  15.  x  =  a,  y  =  a"^, 
z  =  l.  16.  X  =  -  Sa,  2/  =  2a6,  ^  =  -  abc.  17.  3  a6c  =  Sa^. 
18.  X  =  3,  2/  =  1,  2;  =  -  2.         19.   Sa*  +  2  2:a2&2  =  8  ^252. 

d.  1.  x  =  6,  -2xV  2/  =  4,  -  IxV  2.  X  =  i(a  +  \/a*-8  5), 
2/=  1  (a -</«*- 8  6).  3.  x  =  ±50,  y  =  ±16.  4.  x=:J(3±v5), 
2/  =  i(l  ±  V^)-  5._x  =  7,  2/  =  3.  6.  X  =  4,  i(-  H  +  *V59), 
2/  =  6,  i(_  11  _  iV59).  7.  X  =  15,  -  16,  2/  =  9,  -  10.  8.  x  =  i, 
K  V3  -  2),  2/  =  1,  V3  -  2.  9.  X  =  10,  2/  =  8.  10.  x  =  3,  2/  -  2. 
11.  x=  1.786...,  2^  =  1.731....  12.  x  =  -i,y=:-j\.  13.  x  =  i, 
2/  =  i.  14.  x  =  y/{ac/b),  y=^(ab/c),  z=:^(bc/a). 


330  ANSWERS. 

15.  2a/\/2T2V^,  qv^2  +  2V5,  j  q\/(2  +  2 V5)3.  16.  x  r=  3, 
y  =  7,  z  =  9.  n.  x=  Vbc^'^a-^,  y  =  '\/ab^^c-^  z  =  Vca>^h-\ 

18.  27  a^2  =  4(i)  -  2  a)3.  19.  x  =  2,  y  =  4.  20.  x  =  2,  ?/  =  3. 
21.  X  =  4,  ?/  =  6,  ;?  =  3.         22.  15,  36.         23.  15,  20,  25. 

XI. 

a.  1.  999666,  1098010,  0.  2.  0.  3.  -  7617.  4.  0.  5.  4  in 
each  case.      6.0.000211.      7.-0.0114-...      9.  12^2  -  31  x  +  11. 

b.  1.  (x+ l)3-6(x  +  l)2+ll(x  +  l)-5.  2.3  65+30  6* 
+  119  63+238  62  +  249  6  +  106.      3.  x^-l.      4.   (x  +  3)5-(x+3)+l. 

5.  (a  +  6)5-3  6(a  +  6)*  -  4  h^{a  +  6)2  +  6^. 

6.  (x  -  1)3  +  3(x  -  1)2  -  4(x  -  1). 

C.  1.  i.  2  and  3.  ii.  —  2  and  —  3.  iii.  —  3  and  —  2,  1  and  2. 
iv.  Oand-1,  land2.  2.  2.2284.  3.  1+^3.  4.  \+ ^(2-{-i^^). 
6.  2-^17.  7.  3,  J(-l  +  V")-  8.  1.46460....  9.  14.0487  in. 
10.  2.5119.... 

XII. 

a.  1.  i.  m +(n  —  l)(j9  —  w).  ii.  a  +  6(n  —  1)(2  -  n). 
iii.  1(3  n  -  1).  iv.  -^^(n  -  1).  v.  \{Q\  _  3  6  +  n  •  6  -  8). 
vi.  K92-8w).  vii.  K«+^)-  2.  i.  45.  ii.  12i.  iii.  63. 
iv.  w(2a  +  3  6-?i6)/2a(a  +  6).     v.  816.     vi.  n(n+l).     3.  48000. 

4.  \  7i{n  -  1).  5.  8729.  7.  nth  =  i(6  w  -  5).  8.  6  or  12  terms. 
9.  11.  10.  19800.  12.  $1630.  13.  i.  46  days.  ii.  91  days. 
iii.  20  or  71  days.  iv.  no.  14.  i.  27  days.  ii.  7  or  20  days. 
15.  62.  16.  49J  s,  n  =  2  m  +  1.  17.  $  3140.  19.  diff.  ==  ^(b  -  a). 
20.  nth  =  i  71(^  +  1).  21.  wth=rn2.  22.  nth=  n^  23.  i)=4(n-l). 
24.  i{n  +  l)2(w  +  2)2.         25.  J  w(n2  +  1). 

b.  1.  i.  K-^"  -  1).  ii-  lU  -(-  I)"}-  iii-  f  {1  -(A)"}»  f- 
iv.   V2  +  i.    2.  n  +  1.    3.  1{(V2-1)/V2}«-^.    4.  12.056 ...  gals. 

5.  s^^/2'S%  ■j\7rs2.  6.  a26/(2a+6).  7.  10.  8.  iaw(n  +  l) 
+  («"+! -a)/(a-l).         10.  6381407. 

C.   1.   2,  4,  8,  16,  32,  64,  128,  256. 

d.  1.  2a6/(a  +  6).        6.  1,  2,  4. 

e.  1.  $1069.20.  2.  $1156.  3.  $8192,  4.  $118.75. 
5.  $1734.90. 


ANSWERS.  831 


XIII. 

a.  1.  i.  6.    ii.  6.    iii.  840.    iv.  (n-l)\/(n-l  -  m)l     2.  720. 
3.  10.    4.  1260,  120,  90720.     5.  1  in  30.     6.  36.     7.   1  in  72.     8.  60. 

b.  1.  Qa={n-2)b.       2.  nb  =  a(r)\.       3.  385.       7.  3  in  10. 

C.   1.  i.  {n(w  -  1)  •••  (n-r-\-  l)a«-^x^}/r  !.  ii.  i  x  (-  l)*". 

iii.  {2  n(2  n  -  1)  ••.  {2  n  -  r  +  l)xr}/r  !.  3.  ^^CqX^. 

^.  {(n-r)x}/{air  +  l)}.      5.  ^^^^=-^>r >?^^^:^-l.      6.    1080. 

d.  2.  1,  0,  ~1,  accordingas  wisof  formSm,  3m  +  1,  or3m+2. 

3.  coho  +  cihi  +  C2h2  +  ...  +  Cnhn.     5.   -  197.     6.   ^~?"i'^'^~,\^'- 

n  !(2  n  —  1)! 

I        4.9     4.8V9/       4.8.12\9/  / 

e.  2.  1  .3.5...(2r-3)af/2.4.6...2r.      3.  l+^x  +  -^x^ 
x3  4-  -.        4.  a(l-x+  I  x2  -  ix3  -  Jx*).        9.  V^. 


2.4.6 


XIV. 

7.   i.  The  first,     ii.  The  second,    iii.  The  first.     9.  Less  than  4 
for  all  values  of  x  from  1  to  3. 


a.  1. 


XV. 
4  3 


X-2X-1  X-3X-2 

_-t i-+        '       ■ 

2(x-l)      x-2     2(x-3) 

a-\-l  a^  +  b 


(a— 6)(6  — x)      6(a  — 6)(a  — x) 
21  42       ^        14  vi  ^  ^ 


x  +  2     2x  +  3      (x  +  2)2  2(a-x)      2(a  +  x) 

b.   1.  l+x-hx'^  +  x^+-",     2.  l  +  ix  +  fx2-TVa;^+Tl^a:*.-.. 
3.  i_a;+ 1x2-1  a-3+fx*....      4.  l  +  ix-ix24-A^^-]j?%-'^*--- 

6.     (1  +X)-«  =  1  -CiX+  (C12-C2)  X2-  (Ci3-2  C1C2  +  C3)  X3+  (Ci*-3  Ci2C2 


332  ANSWERS. 

+  2ciC3  4-C22-C4)ic*.-..      6.  x=l/ai'y—a2/ai^'y'^-\-(2a2^-asai)/ 

9.  a  =  1,  6  =  6,  c  =  62/2  \,  d  =  h^/^  !,  etc. 

C.   1.  (nH-l)x«-i.      2.   (2-x+2x2 )_(l_2ic+3x2-4x3...). 

3.  Jn(n2  +  2).      4.  J  n(n  +  l)(w;2+ n  +  2).      5.  |  n(w  +  l)(4  w-1). 

6.  4(11 -2w)(l  +  n).        7.  in(n  +  l)(ri  +  2)(n  +  3). 
12 

d.  1.  a^h  =  4  c.  3.  (ca'  -  c'aY  =  {ah'  -  a^h)  (be'  -h'c). 

4.  \  =  2VAB-{A  +  B).  6.  3.  9.  (b-p^)/2p.  10.  (l+2x) 
/(l_aj-x2).  11.  a -6  =  1  12.  (x2  -  2  x  -  l)2-(x  +  1)2. 
13.   (x  +  2?/  -  l)(x  -  y  +  3).         14.  m  =  f 


XVI. 

a. 

1. 

1,  i  li 

If ,  AV,  if?. 

3.  558, 

552. 

4. 

¥,  m- 

5. 

b. 

1. 

i(Vi5- 

-1). 

2.  K2V39- 

9). 

4. 

V6. 

5.  1  + 

1 

'94- 

2  +  - 
2 

1       1 

+  4  +  "• 

6. 

'-sV 

-.,  4-f 

1      1       1      1      1      1  ' 

2+  1+  3+  1+2+  8  + 

XVII. 


1  ^ 


a.  1.  2,  4,  6,  -  3,  -  5.  3.  1,  i,  1,  23,  2«  6.  i.  1.537.... 
ii.  0.8379  ....  iii.  1.242.  iv.  x  =  2l,y  =  3f.  v.  log( ^2  +  1)/ 
21oga.  vi.  iV(l  +  81ogn/loga)-i.  7.  -^^log2+ilog3-ilog5. 
8.  1.80618,  2.40824,  1.05361,  1.69897,  1.39794.  9.  log2  +  log  3, 
log2  +  21og3,  31og2  +  21og.3,  -(2  log  2  +  log  3),  -21og2, 
-(31og2  +  log3).       10.   8.       11.    «  =  log 2/log  1.05. 

b.  1.  6.7449.  2.  0.0018542.  3.  0.57122.  4.  7.4123.... 
6.    0.69897.       6.    0.93936. 

c.  10.    Se. 

d.  1.  x  =  e«(»+i)-a. 

XVIII. 

a.  1.  i?  =  1  -  2  X  +  x2,  G^  =  1/i?.  2.   i?  =  1  -  2  X  +  x2. 

G^rz:(l  +  x-x2)/jR.     3.    6  x*  +  7  x^.     4.    1  +  X  -  x2  +  3  x^  -  3  X* 
+  5x5-5x6....       6.  (3  +  8x)/(3  +  5x+x2),  l/(l-a;+3x2-8x3), 


ANSWERS.  333 

~4ix4,17a^4.  6.    i{l+(_)«(2n  +  3)},  -50. 

7-  i{l  +  (-)"(!  -  2  n)}.      8.   (3«+i  -  2«+i)x^ 

b.   2.  50  +  -(^2  _  15  ;^  +  26).  3.    1  +  7  n  -  n2. 

4.  -^^(n^  _  4  w2  +  13  n  +  6).  6.  4,  2,  1,  1,  2,  4,  7.  8.  45.14, 
37.89.      9.   -0'.13.       10.   0.41337. 

C.  1.  {l-(n+2)x«+H(w+l)x«+2}/(i_x)2.     2.  (2"+2-n-3)/2« 

3.  {l-i(w  +  l)(w  +  2)x»  +  n(n  +  2)x»+i-|  n(^+_l)a;«+2}/(l-x)8. 

4.  l  +  w.2"+i.  5.  {H-(-)"(7i+2x«+Hw  +  l^"+2)j/(i+aj)2. 
6.  i{l+(-)"(2n  +  3)}.  7.  n/(n+l).  8.  n/(2nH-l), 
9.  7i/3(/i  +  1)  +  n/6(n  +  2)  +  n/9(?i  +  3).  11.  f  12.  J. 
13.   a.       14.    conv.  ifx<l.      15.    x<l.       17.   4.       18.    |. 


XIX. 

a.  1.  i.  -18.  ii.  173.  iii.  abc  +  2  fgh  -  af^  -  hg^  -  ch^. 
iv.  x^.  V.  xz/(y  —  x)(l  —  ic)(l  —  y),  vi.  3  a6c  —  Sa^  vii.  —  4. 
viii.  c(ae  -  c^)  +  6(c(Z  -  6e)  +  (^(^c  -  ad),        2.  0. 

b.  1.  i.  0.        ii.  -  160.        iii.  1. 

C.  3.  $301.  4.  X8  -  X22a  +  A(2a&  -  2/2)  -  (ahc  ■\-2fgh-  ap 
—  hg'^  —  ch'^)  =  0,  wlifTe  ahc  and  fgh  are  collateral  systems  of 
letters.  5.  (a  -  aiy\abi  -  a^h)  =  (b  -  b^y.  6.  mn{BB  -  CQ) 
=  jil(CP-AB)  =  lm(AQ-BP).  7.  m  =  j\(6uo  +  lbui-{-U3-6u2)y 
^  =  T6  (^  ^1  +  9  ^2  —  Wo  —  Wa),  p  =  yV  (^  W3  +  15  W2  +  Wo  —  5  Ui), 
8.    a6c  I a??i?i'  |.     10.  x=—  \  aic^dz  |/|  b\C2,dz  |. 

MISCELLANEOUS. 


2.  151.  3.  -2(1+2x2  +  2x4  +  2x6+...).  4.  x  =  a/\/a234. 
5.  x=a,  y  =  \/a^  z  =  \.  6.  none.  7.  l  +  2(x— x^— x^+x^+x^—). 
8.  60  and  40.  9.  max.  3,  min.  \.  10.  x  must  not  lie  between 
-fandS.  12.1a.  13.  min. -(a- 6)2/4  a?).  14.  (-)"(3  7i-l)x^ 
15.  x=:3,  i/  =  l.  16.  5.  17.  $120.  18.  5,  7,  9.  19.  in(w2  +  2). 
20.  J/(2  n  -  1),  1  n2/.  21.  1,  -  1,  i-  22.  i  5.  23.  1,  3,  9. 
24.  no,  5th.  25.  0.1  per  unit.  26.  ^,  iV»  ^^•>  tui  ij%. 

27.  (x-lly  +  l)(2x  +  ?/-3).         29.  z=j\x+^x^  30.  4. 


334  ANSWERS. 

31.  52.  32.  105.  33.  -1.  34.  593775;  i.  118755;  ii.  20475; 
iii.  17550;    iv.  80730.  35.  -Vr(30r±a°)   seconds  after 

hour  T.         36.  34  hrs.,  at  starting  point.  37.  Put  I  —  z  for  x, 

expand,  and  then  make  z  =  \.  38.  13.  39.  3  +  i     -  .... 

3  +  6 
40.  1027,  715.  ^l.  {ah{h-a)-\-hc{c-h)-\-ca{a-c)}/ahc, 

42.  a-\-b-^c,  43.  w5  +  5  w^  +  lO  i^H  10  wHS  w.  45.  (2x2-a:  +  2)2 
-(2x-l)2.  46.  (2x-2^  +  ;2)(x-3?/-l).  47.  t(l  +  ir.-fa:2 
—  ^jX^  +  ••.).  49.   15  miles  from  one  town.  52.  A,  120  ; 

B,  60  ;    C,  24.  53.  x  =  2  ahc/(ab  —  be -\-  ca),  with  sym.  ex- 

pressions for  y  and  z.  54.  h{B  -}-  b  -{-  VBb).  55.  5|  miles. 

67.  y  =  KVa2-4ax3  -  a)/x'^,         z  =  i(- Vo^r^Tox^  -  a)/x^, 

58.  5/(x-l)2  +  3/(x-  l)_(3x  +  4)/(x2-x+  1). 
61.   {qQ-pP)/{q~p).  62.  greaterpart  =  J  a(V5  -  1). 

63.  2,  3,  4.  64.   Vx  -  2/  +  1  +  Vx  -f  ?/  -  1.  65.  255. 

67.  1  -  i  X.  68.  2  -  X  +  5  x2  -  7  x3  +  17  x^  -  31  x^  +  65  x^  .... 

69.  (2  -  x)/(l  +  x)(l  +  x;^).  71.  z^  -  z^y^  -  y^.  73.  2.8912  in. 
74.  X  -  1,  X  -  2,  X  +  3.  75.  z^-^zy'^^b  ?/^.  76.  437  x  +  557. 
78.  (X  +  5)  {x  +  i  «(1  -  V5)}  .  {x  +  1  a(l  +  yS)}.  81.  i(v'X>2_c2 
4.  Voji  _  ci^,  83.  22  7r/12,  and  18  7r/12.  84.  3.82.  85.  0, 
a(Vr=r^-\/l  +  a2)/(l  +  irt2_Vl-  «4).  86.  22fcu.  ft. 

87.  4  ±  Vli,  -l±L  88.  7i2  z=  c\P  +  m2).  89.  25(7jf  3^5) 
sq.  rds.  90.  x  =  J  a,  a  min.  91.  yx  =  7,  -  8,  ^"^185  —  1). 
93.  i2pn-p-q)/(p-q').  94.  md.  =  d' ^(a'^-^id^'^)/ (d-d'y, 
95.  w=^±aV2s2-a2)/(s2_^2).  93.  18,  or -24.  ^01.  iV799. 
104.    V39i:.3.  105.   14.234  by  20.234.  106.    V2761,  V4561, 

\/388l,  V568I.  108.  x  =(- 3  ±  V21)V  V^-  109-  4v3  +  f7r. 
112.  (1  -x  +  x.l  +  x3)/(l  +  x).  113.  x=(c2  +  2ac-a2)/2c; 
ratio  is  2.         114.  side  :  base  =  13  :  10.         115.  90°.  117.  rate 

=  mmi(h  +  hi)/(mhi  —  mih);  dist.  —  mmihhi(m  +  mi)(Ji  +  ^1) 
/(mhi-mihy.  IIS.  x^-]-y^  =  ay.  119.  12  solutions,  1  +  3 p,  12 -p, 
10+5/).  120.  i(a  +  6±Va2TP).  121.  2^-i+l-+...y 
123.  85  and  118  ;  +  yi^th,  nearly. 


INDEX. 


The  Numbers  refer  to  the  Articles, 


ART. 

Algebraic  expression 7 

Algebraic  subtraction 24 

A Igebraic  sum 25 

Annuity 173 

Annuity  in  reversion 174 

Annuity  foreborne 175 

Antilogarithmic  series 222 

Arbitrary  multipliers 135,  140 

Arithmetic  mean 165 

Binomial  theorem 186 

Characteristic 215 

Coefficient 19 

Coefficients  of  (a+a;)« 31 

Combination 177 

Commutative  law 7, 13 

Complementary  combination 183 

Complex  number 53 

Composite  number 69 

Concrete  quantity 102 

Constant 18 

Convergent 203 

Convergency  of  series 243 

Cube  root  of  1 57 

Cube  root  of  a  number 91 

Cyclic  substitution 25 

Degree 18 

Determinant 253 

Difference  series 234 

Dimension 18 

Directed  segment 107 

Discriminant 200 

Distributive  law 15 

EliramanJ; 138 


ART. 

Euler's  proof  of  binomial  theorem  190 

Equation 22 

Expansion  of  homogeneous  sym- 
metrical functions 30 

Exponent 16 

Exponential  series 222 

Factor 12 

Factorial 179 

Fraction 71 

Function 32 

G.  C.  M.,H.  C.  F 61 

Grenerati  ng  function 229 

Geometric  mean 169 

Graph. , 116 

Graph  of  quadratic 121 

Highest  common  factor 61 

Homogeneous 19 

Identity 22 

Imaginary  quantity 48 

Imaginary  unit 52 

Incompatible  equations 138 

Indefinite  solution 77 

Independent  equations 137 

Indeterminate  equations 129 

Index 16 

Index  law 16,  38 

Infinite  series 35 

Infinity,  qo 75 

Integral  function 33 

Interpolation 236 

Interpretation  of  "fl^. 189 

Irrational  quantity 48 

Irrational  equation 128 

335 


336 


INDEX. 


ART. 

Law  of  signs 14 

Least  common  multiple 66 

Linear  expression 19 

Line  symbol 107 

Linear  factor 42 

Limit 168,  238 

Logarithm 211 

Logarithmic  series 223 

Mantissa 215,  216 

Matrix 255 

Minimum  and  maximum 122 

Napierian  base 211,  220 

Negative  number 8 

Operative  symbol 4 

Partial  fractions 196 

Partial  quotient 203 

Permutation 177 

Periodic  continued  fractions 209 

Perpetuity 173 

Prime  number 69 

Progressions 156 

Proportion 82 

Quadratic,  The 118 

Quadratic  factor 42 

Quadratic  surds,  Theorems  on. . .  99 

Quantitative  symbolfl 4 


ART. 

Ratio 79 

Rationalizing  factor 94 

Real  quantity 48 

Recurring  series 229 

Redundant  equation 138 

Remainder  theorem 149 

Roots 58,  117 

Rule  of  Sarrus 259 

Scale  of  relation 229 

Sigma  notation 28 

Similar  surds 92 

Simultaneous  equations 134,  136 

Solution 23,  59,  134 

Special  roots 76 

Summation  of  series 199 

Sum  of  a  series 238 

Surd 88 

Symbolic  geometry 106 

Symmetry 26,  78 

Synthetic  division 39 

Term 10 

Term,  nth 157,  228 

Undetermined  coeflScients 195 

Unit-variable 114 

Variable 13 

Verbal  symbols 4 


THE    PRINCIPLES 

OF 

ELEMENTARY  ALGEBRA, 

BY 

NATHAN  F.   DUPUIS,  M.A.,  F.E.S.C, 

CB  in  the  University 
Canada. 

i2mo.    $1.10. 


Professor  of  Pure  Mathematics  in  the  University  of  Queen's  College,  Kingston, 
Canada. 


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The  whole  covers  pretty  well  the  whole  range  of  elementary  algebraic 
subjects,  and  in  the  treatment  of  these  subjects  fundamental  principles 
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book on  the  subject  I  am  acquainted  with.  It  is  up  to  the  time,  and 
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to  have  seen  a  better.  .  .  .  It  is  the  outcome  of  a  long  experience  of 
school  teaching,  and  so  is  a  thoroughly  practical  book.  .  .  .  Buy  or 
borrow  the  book  for  yourselves  and  judge,  or  write  a  better.  ...  A 
higher  text-book  is  on  its  way.  This  occupies  sufficient  ground  for  the 
generality  of  boys."  —  Academy, 

HIGHER  ALGEBRA.  A  Sequel  to  Elementary  Alge- 
bra for  Schools.  By  H.  S.  Hall,  M.A.,  and  S.  R.  Knight,  B.A. 
Fourth  edition,  containing  a  collection  of  three  hundred  Miscella- 
neous Examples  which  will  be  found  useful  for  advanced  students. 
12mo.    il.90. 

OPINIONS  OF  THE  PRESS. 
**  The  *  Elementary  Algebra '  by  the  same  authors,  which  has  already 
reached  a  sixth  edition,  is  a  work  of  such  exceptional  merit  that  those 
acquainted  with  it  will  form  high  expectations  of  the  sequel  to  it  now 
issued.  Nor  will  they  be  disappointed.  Of  the  authors'  '  Higher  Alge- 
bra,' as  of  their  '  Elementary  Algebra,'  loe  unhesitatingly  assert  that 
it  is  by  far  the  best  work  of  its  kind  with  which  we  are  acquainted. 
It  supplies  a  want  much  felt  by  teachers." —  The  Athenxum. 

"  .  .  .  Is  as  admirably  adapted  for  college  students  as  its  predecessor 
was  for  schools.  .  .  .  The  book  is  almost  indispensable  and  will  be 
found  to  improve  upon  acquaintance."  —  The  Academy. 

"...  The  authors  have  certainly  added  to  their  already  high  repu- 
tation as  writers  of  mathematical  text-books  by  the  work  now  under 
notice,  which  is  remarkable  for  clearness,  accuracy,  and  thorough- 
ness. .  .  .  Although  we  have  referred  to  it  on  many  points,  in  no 
single  instance  have  we  found  it  wanting." —  The  School  Guardian, 


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ALGEBRAICAL     EXERCISES    AND    EXAMINATION 

Papers.  By  H.  S.  Hall,  M.A.,  and  S.  R.  Knight,  B.A.  16mo. 
Cloth.     GO  cents. 

This  book  has  been  compiled  as  a  suitable  companion  to  the  **  Ele- 
mentary Algebra"  by  the  same  Authors.  It  consists  of  one  hundred 
and  twenty  Progressive  Miscellaneous  Exercises  followed  by  a  compre- 
hensive collection  of  Papers  set  at  recent  examinations. 

ARITHMETICAL    EXERCISES    AND    EXAMINATION 

Papers.    With  an  Appendix  containing  questions  in  Logarithms 
and  Mensuration.    By  H.  S.  Hall,  M.A.,  and  S.  R.  Knight,  B.A 
Second  edition.     IGmo.    60  cents. 

KEY  TO  THE  ELEMENTARY  ALGEBRA  FOR  SCHOOLS. 

12mo.    $2.25. 

KEY  TO   THE    HIGHER    ALGEBRA    FOR    SCHOOLS. 

12mo.    $2.60. 

iit*^  These  Keys  are  sold  only  upon  a  teacher's  written  order. 

HALL  and  STEVENS.  —  A  Text-Book  of  Euclid's  Ele- 

MENTS.  Including  Alternative  Proofs,  together  with  additional 
Theorems  and  Exercises,  classified  and  arranged.  By*H.  S.  Hall, 
M.A.,  and  F.  H.  Stevens,  M.A. 

Books  I.-VI.    Globe  8vo.    $1.10. 

Also  sold  separately : 

Book  I.    30  cents.  Books  I.  and  II.    50  cents. 

Books  I.-IV.    75  cents.  Books  III.-VI.    75  cents. 

Book  XI.    30  cents.  Books  V.,  VI.,  XI.   70  cents. 

*'  This  is  a  good  Euclid.  The  text  has  been  carefully  revised,  the  defini- 
tions simplified,  and  examples  added  illustrative  of  the  several  propo 
sitions.    We  predict  for  it  a  very  favorable  reception."  —  Academy. 

LEVETT  and  DAVISON.  — The  Elements  of  Trigonom- 

etry.  By  Rawdon  Levett  and  A.  F.  Davison,  Masters  at  King 
Edward's  School,  Birmingham.    Crown  8vo.    $1.60. 

This  book  is  intended  to  be  a  very  easy  one  for  beginners,  all  diffi- 
culties connected  with  the  application  of  algebraic  signs  to  geometry 
and  with  the  circular  measure  of  angles  being  excluded  from  Part  I. 
Part  II.  deals  with  the  real  algebraical  quantity,  and  gives  a  fairly 
complete  treatment  and  theory  of  tlie  circular  and  hyperbolic  functions 
considered  geometrically.  In  Part  III.  complex  numbers  are  dealt  with 
geometrically,  and  the  writers  have  tried  to  present  much  of  De  Mor- 
gan's teaching  in  as  simple  a  form  as  possible. 


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66  FIFTH  AVENUE,  NEW  YORK. 


ARITHMETIC  FOR  SCHOOLS.     By  Rev.  J.  B.  Lock, 

Fellow  and  Bursar  of  Gonville  and  Gains  College,  Cambridge ;  for- 
merly Master  at  Eton.  Third  Edition,  revised.  Adapted  to  Ameri- 
can Schools  by  Prof.  Charlotte  A.  Scott,  Bryn  Mawr  College, 
Pa.    70  cents. 

The  Academy  says:  "'Arithmetic  for  Schools,' by  the  Rev.  J.  B. 
Lock,  is  one  of  those  works  of  which  we  have  before  noticed  excellent 
examples,  written  by  men  who  have  acquired  their  power  of  present- 
ing mathematical  subjects  in  a  clear  light  to  boys  by  actual  teaching 
in  schools.  Of  all  the  works  which  our  author  has  now  written,  we^ 
are  inclined  to  think  this  the  best.'* 

TRIGONOMETRY  FOR  BEGINNERS,  AS  FAR  AS 

THE  Solution  of  Triangles.  By  Rev.  J.  B.  Lock.  Third  Edi- 
tion.   75  cents.    Key,  supplied  on  a  teacher's  order  only,  ^1.75. 

The  Schoolmaster  says :  ''  It  is  exactly  the  book  to  place  in  the  hands 
of  beginners. .  .  .  Science  teachers  engaged  in  this  particular  branch 
of  study  will'  find  the  book  most  serviceable,  while  it  will  be  equally 
useful  to  the  private  student." 


TO  BE   PUBLISHED    SHORTLY. 
INTRODUCTORY    MODERN     GEOMETRY     OF     THE 

Point,  Ray,  and  Circle.  By  William  B.  Smith,  Ph.D.,  Profes- 
sor of  Mathematics  in  Missouri  State  University. 

THE  ELEMENTS  OF  GRAPHICAL  STATICS.    A  Text- 

Book  for  Students  of  Engineering.  By  Leander  M.  Hoskins, 
C.E.,  M.S.,  Professor  of  Theoretical  and  Applied  Mechanics,  Univer- 
sity of  Wisconsin. 


Correspondence  from  professors  and  teachers  of  mathematics,  re- 
garding  specimen  copies  and  terms  for  introduction,  respectfully 
invited* 


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14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPTT 

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